ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012) 72 ndash 82

DOI 103813AAA918493

Calculation of Head-Related Transfer Functionsfor Arbitrary Field Points Using SphericalHarmonics Decomposition

Martin Pollow1) Khoa-Van Nguyen23) Olivier Warusfel2) Thibaut Carpentier2)Markus Muumlller-Trapet1) Michael Vorlaumlnder1) Markus Noisternig2)

1) Institute of Technical Acoustics RWTH Aachen University Germany mpoakustikrwth-aachende2) Acoustic and Cognitive Spaces Research Group IRCAM ndash CNRS UMR STMS Paris France3) sonic emotion labs Paris France

SummaryThe head-related transfer function (HRTF) characterizes the transfer path of sound from a certain position inspace to the ear of a listener Traditionally HRTFs have been measured for relatively distant sources at dis-crete positions on a surrounding sphere It is well known from literature that the HRTF changes significantlyfor sources in the proximal region ie at distances less than one meter from the head In practice the mea-surement of near-field HRTFs requires a significant number of measurement points at different distances fromthe head it may thus not be feasible for many projects An essential question is whether HRTFs can be calcu-lated for any arbitrary position in space from measurements on a single radius HRTF data can be representedin the spherical wave spectral domain thus turning the range extrapolation problem into an acoustic radiationproblem The range extrapolation is then calculated using the corresponding wave propagation terms This pa-per presents a comprehensive study of the spherical acoustic method for calculating HRTFs at arbitrary fieldpoints Near-field HRTFs are calculated for two dummy heads and then compared to other range extrapolationmethods known from literature Particular attention is paid to the effect of incomplete spherical data sets on thereproduction accuracy and different regularization methods are discussed The results are compared to near-fieldmeasurements for these dummy heads and numerical boundary-element method (BEM) simulations thereof Thespherical acoustic method shows a good agreement with measurement data thus providing an efficient methodfor near-field binaural synthesis using already existing HRTF data sets

PACS no 4360Ek 4366Pn

1 Introduction

Human beings are able to localize sound with a remarkableaccuracy Psychophysical studies have shown that the hu-man auditory system uses different mechanisms for soundlocalization In the horizontal plane the angular direc-tion of sound is basically determined from interaural timedifferences (ITD) and interaural level differences (ILD)whereas sound elevation mainly depends on direction-dependent spectral cues generated by diffraction and scat-tering effects of the pinna head and torso [1] These spec-tral cues also provide information that helps resolving theambiguities inherent in ITD and ILD cues [2] The rela-tionship between the position of a sound source in three-dimensional space and the sound pressure generated bythat source at the entrance of the ear canal can be rep-resented by a linear spatial filter the head-related trans-fer function (HRTF) [3] Processing an audio signal witha set of HRTF filters and playing the output signals back

Received 28 May 2011accepted 22 August 2011

over headphones thus creates the illusion of a virtual soundsource at the corresponding position in space

HRTFs are typically measured for a large number of po-sitions on a surrounding sphere in the far-field of the head[4 5 6 7] They can also be computed by numerical sim-ulations solving the wave equation subject to boundaryconditions on the headrsquos surface [8 9 10 11 12] Bothare nontrivial and time consuming tasks and thus exceedthe means of many projects Moreover individual differ-ences in the human anatomy require repeating this proce-dure for each individual subject in order to obtain percep-tually convincing results [13] In this study all measure-ments were performed on dummy heads without the needto expose subjects to exhaustive measurements Dummyheads provide a well defined geometry and surface mate-rial structure and are therefore particularly suited for nu-merical simulations Furthermore only the quality of re-construction and interpolation has been evaluated and notthe quality of the HRTF itself It is thus sufficient to testthe algorithms on dummy heads even though HRTFs varyconsiderably between individuals

72 copy S Hirzel Verlag middot EAA

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

The spatial resolution of measured HRTF data is gener-ally lower than the minimum audible angle of human audi-tory perception cf [1] To provide the listener with a highspatial resolution interpolation techniques have to be ap-plied Angular interpolation methods can be categorizedinto local interpolation only using neighboring HRTFs[14 15] and global interpolation based on the analysisand decomposition of the entire set of measured HRTFsKistler et al [16] and Chen et al [17] proposed the useof principal component analysis (PCA) for angular inter-polation Evans et al [18] first applied the spherical har-monics transform using a Gaussian quadrature method todetermine the optimal measurement points on the sphereThe latter method was further extended to irregular andopen spatial sampling grids using appropriate regulariza-tion methods [19] It was shown that the re-synthesizedHRTFs are in good agreement with the measured data

Human auditory distance perception is less accuratethan the ability to determine the angular direction of asound The perceived distance of a sound source in aroom mainly depends on the direct-sound-to-reverb ra-tio and the distance-dependent level attenuation and high-frequency damping These far-field effects are commonlyapplied to create distance perception in virtual audio en-vironments [20] In the proximity of the head auditorydistance perception appears to be determined by differentfactors Brungart and Rabinowitz [21] studied the phys-ical and perceptual differences of near-field and far-fieldHRTFs They have shown that the HRTF changes sig-nificantly for sources at distances less than 1 m from thehead More recently Lentz et al [22] perceptually evalu-ated measured HRTFs at different distances from the headshowing limits of noticeable differences between near-field and far-field HRTFs Borszlig [23] used a computation-ally efficient analytical model for the simulation of near-field sources which takes both the head-related and source-related near-field effects into account

In the proximal region HRTFs have to be measuredon a dense angular mesh at different distances from thehead which requires a significant number of measurementpoints Thus a crucial question is whether this data canbe calculated from far-field measurements at a single dis-tance Romblom and Cook [24] proposed near-field com-pensation filters and a geometric HRTF selection approachto simulate nearby sources Duraiswami et al [25] andZhang et al [26] extended the work of Evans et al [18] byfurther applying the radial wave propagation terms trans-forming the range extrapolation of HRTFs into an acousticradiation problem They evaluated simulation results on aspherical head model [27]

This article is organized as followed Section 2 reviewsthe spherical wave spectrum and range extrapolation inthe spherical harmonic domain It further introduces thenormalized correlation of spherical functions as a qualitymeasure for the simulation results HRTFs are typicallyderived from measurements over an incomplete portionof the sphere as the source directions in the lower hemi-sphere intersect with the subjectrsquos body Several strategies

are possible to obtain the spherical wave spectrum fromthese incomplete data sets The problems of matrix inver-sion and regularization are briefly discussed in this sectionIn section 3 range extrapolated HRTFs for two differentdummy heads are compared to measurement data in theproximal region and boundary-element simulation (BEM)results Section 4 analyzes the results and further comparesthe range extrapolation in the spherical wave spectral do-main with other range extrapolation methods proposed inliterature It is shown that the spherical acoustic methodprovides one efficient means of near-field binaural synthe-sis with a good agreement with measurement data

2 Interpolation and range extrapolation ofHRTFs

Applying the principle of reciprocity HRTFs can be re-formulated as an acoustic radiation problem [28] Whenthe volume velocity at the entrance of the ear canal isknown the radiated pressure field is completely definedoutside a region that encloses all contributing scatteringsources It can be calculated from the Helmholtz equationassuming that the Sommerfeld radiation condition is ful-filled [25 28]

21 Spherical wave spectrum

The solution of the wave equation in spherical coordinatesunder the assumption that all inward travelling waves arenegligible results in the expansion of the acoustic pressurefield

p(r θ φ k) =+infin

n=0

n

m=minusnanm(r k)Y m

n (θ φ) (1)

at position (r θ φ) and wavenumber k = 2πfc Hereby

Y mn (θ φ) = (minus1)m

(2n + 1)4π

(n minus |m|)(n + |m|)

middot P |m|n (cos θ)eimφ (2)

denotes the complex spherical harmonic function of ordern and degree m and

anm(r k) = bnm(k)hn(kr) (3)

the spherical expansion coefficients hn is the sphericalHankel function of the first kind which is associated withthe outgoing component of the sound field and P

|m|n are

the associated Legendre functions bnm denotes the spher-ical expansion coefficients which have to be determinedby a projection-based analysis

One can note from equations (1) and (3) that for a givenwave number k the sound pressure field is entirely de-termined by the spherical harmonic expansion coefficientsbnm(k) The prediction of HRTFs is thus realized in twoprocessing steps (a) the expansion coefficients anm(r0 k)are determined by expanding the set of measured HRTFs

73

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

into spherical harmonics on a sphere with radius r0 and(b) the HRTF at a target radius r1 is calculated by rangeextrapolation from radius r0 applying the correspondingspherical Hankel functions

In practice the outer summation in equation (1) is lim-ited to orders n le N which corresponds to a spatial bandlimitation It can be further written in matrix form as

p = Hb (4)

with p denoting the sound pressure vector of Np measure-ment points on a spherical shell and H the Np times (N + 1)2

matrix of spherical harmonics multiplied with the corre-sponding Hankel functions both limited to a maximumorder of N

The measured data at a known distance r0 allows toevaluate the HRTFs at any angular position by means ofthe spherical harmonics expansion with the coefficientsa(r0 k) given in equation (3)

The reproduction accuracy is limited by the order trun-cation as the spatial band limitation yields spatial alias-ing The range extrapolation problem can be solved by ap-plying the corresponding radial propagation functions tocompute the spherical harmonic coefficients anm(r1 k) attarget distance r1

anm(r1 k) = anm(r0 k)hn(kr1)hn(kr0)

(5)

and inserting these coefficients in equation (1) In order toavoid artifacts due to the exponential growth of the spher-ical Hankel functions for higher orders and small argu-ments kr the truncation order is chosen as N = krmin

with rmin being the radius of the smallest sphere enclosingthe listenerrsquos head with all its significant scattering sources[25]

22 Matrix inversion

The spherical harmonics decomposition which solvesequation (4) for the spherical harmonics coefficients b re-quires the inversion of matrix H Depending on the distri-bution of measurement points on the sphere this inversionproblem can be ill-posed ie the solution lacks stabilitywith respect to data perturbations Zotkin et al [19] deter-mined the most appropriate solving methods for differentsampling grids Gaussian irregularly sampled open andclosed grids have been studied

In this study non-Gaussian open grids (with a polargap) were used for most of the HRTF measurements Asa consequence the inversion must incorporate some sortof regularization In general regularization approximatesa solution and balances the approximation error versus thedata error for instance the solution becomes less sensitiveto data perturbations than the least-squares method [29]

A least-squares method with Tikhonov regularizationwas applied to calculate the expansion coefficients fromequation (4) as

breg = HHWH + εD minus1 HHWp (6)

where W = diagw is the Np times Np diagonal matrixof weighting coefficients and D the applied regularizationmatrix The weighting coefficients were obtained from theVoronoi surfaces associated with each measurement point[30] the regularization coefficient ε controls the amountof regularization and is usually set to small values [25]

Standard Tikhonov regularization Using standard Tik-honov regularization the regularization matrix D is chosento be the identity matrix of size (N + 1)2 times (N + 1)2

D = I (7)

Decomposition-order dependent Tikhonov regularizationDuraiswami et al [25] proposed a decomposition-orderdependent diagonal regularization matrix D

D = (1 + n(n + 1))I (8)

where I denotes the identity matrix and n the order of thespherical harmonics of the columns of matrix H

23 Correlation of spherical functions

In order to analyze the performance of the applied rangeextrapolation methods the correlation of two sphericalfunctions f and g can be used as a measure for their simi-larity It is defined as

C(f g) =S2

f (θ φ) g(θ φ) dΩ (9)

with the overbar denoting the complex conjugate cf [31]By transforming f and g into the spherical harmonic do-main the orthogonality of the spherical harmonics can beexploited Equation (9) thus writes as

C(f g) =infin

n=0

n

m=minusnfnmgnm (10)

The normalized correlation in the spherical harmonicdomain C(f g) which is defined as

C(f g) =C(f g)

EfEg

(11)

provides a good measure for the similarity of the sphericalshape of two functions It thus allows to compare spheri-cal functions independent of a possible gain mismatch Ef

and Eg denote the energies of the spherical functions fand g respectively

Using vector notation f = vecfnm the normalized cor-relation can be further expressed as

C(f g) =fHg

|f| middot |g| (12)

with || = 2 being the 2-norm of the coefficient vectorscf [32]

74

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

Figure 1 Geometries of the dummy heads used for simulationsand measurements a Dummy head 1 b Dummy head 2

3 HRTF measurement and numerical sim-ulation

For this study two different dummy heads which are de-picted in Figure 1 were used for both simulations andmeasurements The first herein referred to as ldquodummyhead 1rdquo is a HEAD acoustics HSU III mannequin thatprovides a relatively simple head torso and ear geome-try it is thus particularly suited for numerical simulationsThe second herein referred to as ldquodummy head 2rdquo is acustom-made mannequin produced at ITA Aachen with asimple torso but a much more detailed ear geometry [33]

31 Boundary-element method simulation

In order to obtain noise-free data for further comparisonsboundary-element-method (BEM) simulations were usedto calculate the HRTFs for the two dummy heads Thephysical structure of the heads was modeled using lin-ear triangular elements and the reciprocal acoustic methodwas applied for calculating the sound field around thehead propagation through the head structure was ignoredThe boundary condition can be defined by applying a con-stant surface velocity to the elements at the entrance ofthe ear canal on one side of the head With this boundarycondition the acoustic radiation was simulated to differ-ent field point grids at various distances from the centerof the head To calculate the directional transfer functions(DTFs) from the HRTFs the results of the BEM simulationwere referenced to the pressure created by a point sourceof equal volume velocity at the same radial distance TheDTFs are thus equal to the parts of the HRTFs whichvary for different directions of sound incidence while anyglobal frequency dependency for all directions of soundincidence is eliminated [16]

The LMS VirtualLab 9-SL3 software was used for BEMsimulations for this study the indirect frequency-domainBEM formulation was applied to all simulations

Dummy head 1 HRTFs for the first dummy head werecalculated for frequencies from 20 Hz to 6 kHz at 20 Hzsteps The field point grids are equiangular with an angu-lar resolution of 5 for both elevation and azimuth anglewith distances of 30 50 100 and 200 cm from the headThe simulation of the closest range contains some evalua-tion points that lie within the structure of the dummy head

and thus do not yield valid simulation results These pointswere omitted in the analysis

Dummy head 2 HRTFs for the second dummy headwere calculated for a frequency range from 20 Hz to 5 kHzwith a linear resolution of 20 Hz and from 5 kHz to themaximum frequency of 16 kHz at 50 Hz steps The sizeof the boundary elements directly relates to the upper fre-quency bound for which the calculation is valid A densecomputational mesh results in large BEM matrices andslows down the calculation speed Two meshes with differ-ent spatial discretization were applied to the simulationsthe first a coarse mesh for frequencies up to 10 kHz andthe second a fine mesh for frequencies above this limit

The field mesh was generated with a Gaussian samplingscheme of order 70 for the same distances from the headas for dummy head 1 This results in a high spatial reso-lution of approximately 2 in both elevation and azimuthThe calculation of the spherical wave spectrum was lim-ited to the order 35 and could thus be computed on a sin-gle workstation At the closest range of 30 cm some of themesh points lie inside the dummy head and are thus omit-ted from further calculations

32 Near-field HRTF measurement

For further comparisons HRTFs for both dummy headswere measured on a dense angular grid at several dis-tances from the head The different measurement setupsand methods are described in the following sections

Dummy head 1 ndash Outward (reciprocal) measurementThe HRTF data for dummy head 1 was measured inan anechoic chamber (59 m times 44 m times 425 m) at IR-CAM This anechoic room has a lower frequency boundof 75 Hz The measurement system consists of a Bruel ampKjaer turntable and a custom-built remote-controlled ro-tating arm that can be equipped with loudspeakers andormicrophones

The dummy head was positioned in the horizontal planeusing two laser beams for aligning the interaural axis withthe measurement device For HRTF measurements theacoustic reciprocity method [34] was applied The mea-surement signal was emitted from a Knowles ED29689miniature loudspeaker positioned at the entrance of the earcanal and recorded by a microphone array Eight MonacorMCE 2000 omnidirectional microphones were mountedon a 170 cm rod perpendicular to the mechanical armpointing towards the center of the dummy head The dis-tances between the microphones and the ipsilateral ear onthe ear axis were set to 20 30 50 70 100 136 170 and200 cm All microphones were powered and amplified bya STUDER LMS Carouso Mic24 ADAT preamplifier andsignals were played back and recorded with a RME Digi-face HDSP Hammerfall audio card with 48 kHz samplingrate The excitation signal was an exponential sweep [35]with a length of N = 216 samples A total number of 865impulse responses was measured for each radial distanceproviding sampling points from -30 to 90 elevation and

75

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

360 azimuth with a step size of 10 in elevation and 5

in azimuthAll microphones had been equalized to compensate for

temporal phase and level differences The closest micro-phone position at full lateralization ie on the interauralaxis was defined as global time reference point Diffuse-field equalization filters as described in [36 37] were ap-plied to all measurements These filters are derived froma weighted average over all measurements for one micro-phone at a known distance The applied weighting coeffi-cients correspond to the Voronoi surfaces [30] associatedwith each measurement position After the diffuse-fieldequalization directional transfer functions (DTFs) wereobtained for further calculations

Dummy head 2 ndash Inward measurement The near-fieldHRTFs for dummy head 2 were measured by Lentz [38]in a semi-anechoic chamber (11 m times 6 m times 5 m) at ITAAachen This anechoic room provides a lower cut-off fre-quency of approx 100 Hz Similarly to the setup used atIRCAM a computerized positioning system consisting ofa turntable and a rotary arm for a moving sound sourcewere used that allows to measure HRTFs in the proximalregion of the head

The dummy head was mounted and aligned on theturntable and a pair of microphones was positioned at theentrance of the ear canals using custom made earmoldshells The entire measurement equipment was calibratedand referenced to the measured sound pressure of a mi-crophone in absence of the dummy head yielding the di-rectional transfer functions (DTFs) Due to the solid floorin the semi-anechoic room a strong first ground reflectionoccurred which was eliminated by extracting the first partof the measured impulse response

An equiangular spatial sampling grid with an angularresolution of 5 in elevation and azimuthal direction wasapplied which results in 2664 measurement positions onthe full sphere In practice for source distances close tothe head only partial sphere measurements could be ob-tained Thus the coverage of the HRTF data varies withthe measurement distance Whereas for distances at 20 and30 cm only the upper hemispherical HRTF data could beobtained the measurements at distances 40 50 75 100and 200 cm cover larger parts of the sphere encompassingall elevations from +90 (above the head) to -45 -55-75 -75 and -90 respectively

A more detailed description of the measurement setupcan be found in [38]

4 Analysis of the results

In this section the results of the proposed methods aregiven First the correlation analysis is used to show thematch of original and extrapolated HRTFs as functions offrequency This provides the possibility to study the ef-fect of regularization on partially defined data Secondthe HRTFs in the horizontal plane are plotted over fre-quency which allows to observe the fine structure of origi-nal and calculated functions in the spatial domain Finally

1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 2 Correlation between measured HRTFs and extrapo-lation results from measurement data obtained at a differentrange (dummy head 1) a Calculation without regularization(weighted least squares) b Calculation using decomposition-order dependent regularization with ε = 10minus8

the given calculation using the spherical wave spectrum iscompared to alternative near-field compensation methodsfound in literature

41 Correlation analysis of the HRTF extrapolation

The normalized correlation function C(f g) as defined inequation (12) is used to evaluate the range extrapolationof HRTFs in the spherical wave spectral domain Therebythe lower bound C = 0 and the upper bound C = 1 cor-respond to a total mismatch and a perfect match of thesimulated and extrapolated data respectively

Measurement data The correlation over frequency formeasurement data of dummy head 1 is depicted in Fig-ure 2 As mentioned earlier diffuse field normalizationwas applied to all data sets before correlation the resultsthus refer to the directional transfer functions (DTFs) Asa result of missing measurement data around the southpole ie the polar gap the weighted least squares ap-proach fails To obtain accurate range extrapolation data inboth outward and inward directions regularization meth-ods have been applied The regularization matrix is givenin equation (8) and the regularization parameter was set toε = 10minus8 In doing so the high frequency part of the ana-lyzed HRTFs could be more accurately reproduced thanwhen using ε = 10minus6 as has been suggested in [25]As already mentioned in Sec 21 the maximum sphericalharmonic order was truncated to N = krmin see also[25] rmin was set to 30 cm and thus encloses all scatteringsources

The correlation measure shows similar results for boththe inward and the outward extrapolation of HRTFs Therelatively high correlation does not reflect differences inthe fine structure of the HRTFs and their dependencies onthe direction of range extrapolation This will be discussedin more detail in Sec 42 The correlation curves for bothinward and outward extrapolation follow the same trend

76

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

and show a drop above 6 kHz which might be caused bymeasurement uncertainties

BEM simulation data Numerical simulations allow toanalyze the HRTF range extrapolation and interpolationmethods without the effects of measurement noise For thisreason the BEM method described in Sec 31 was appliedto dummy head 1 The simulation results were truncatedto elevations from+90 (north pole) to -30 and thus coverthe same portion of the sphere as the measured HRTFs

Figure 3 depicts the correlation between range extrapo-lated and BEM simulated HRTFs derived from complete(full sphere) and incomplete (partial sphere) data sets andfor different inversion methods The range extrapolatedHRTFs are in a good agreement with the BEM simulationresults and show in general slightly higher accuracy fornear-field to far-field range extrapolation than vice versa

It can be seen that the correlation monotonically de-creases with increasing frequencies up to 180 Hz This isbecause the spherical harmonic of order zero (monopole)fails to model the spatial variations of the HRTF dataAbove this frequency the maximum order is set to one(monopole plus dipole) and the extrapolated HRTFs areagain well correlated with the BEM simulations Thiseffect can be minimized by introducing smooth (order-weighted) transitions or by increasing the maximum orderin this frequency region However due to the high corre-lation values (above 095) this has not been investigatedfurther in this study as it is unlikely to cause significantaudible artifacts

The correlation curves in Figures 3b and c illustrate theimportance of regularization for data from measurementsover an incomplete portion of the sphere Figure 3b clearlyshows that weighted least squares inversion fails for thepartial sphere at frequencies above 15 kHz with the cor-relation decreasing rapidly for higher frequencies In Fig-ure 3c Tikhonov regularization with a regularization pa-rameter ε = 10minus4 was applied and the extrapolated HRTFsare well correlated with the BEM simulated data over theentire frequency range It is important to note that the BEMsimulations were limited to frequencies up to 6 kHz How-ever measurement results for the same dummy head asdepicted in Figure 2 show a good agreement of range ex-trapolated and measured HRTFs over almost the entire fre-quency range of human hearing (up to 16 kHz) when reg-ularization is applied

42 Azimuthal frequency plots for outward and in-ward range extrapolation

In this section the results of inward and outward range ex-trapolation are compared graphically Only the azimuthalvariation (horizontal plane of the HRTFs) is depicted asa magnitude plot over frequency Hereby the radiation isnormalized to the average (omnidirectional) radiation foreach frequency each plot thus denotes the deviation fromthe average as level values in decibels

For all dummy heads the results for the left ear were an-alyzed each of the plots in Figures 4 to 8 thus show their

20 40 60 100 200 400 1k 2k 4k 6k09

095

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

(c)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 3 Correlation between extrapolated HRTFs and corre-sponding BEM simulation results for dummy head 1 at dis-tances 200 cm (solid line) and 30 cm (dashed line) The range ex-trapolation is calculated from BEM simulated HRTF data whichcovers (a) the full sphere or (b)ndash(c) only a partial sphere witha polar gap below minus30 elevation (with and without regulariza-tion) Thus ldquo30 cmrarr200 cm denotes range extrapolation fromradius 30 cm to 200 cm compared to BEM simulation results atradius 200 cm a Full sphere without regularization (weightedleast squares) b Partial sphere without regularization (weightedleast squares) c Partial sphere standard Tikhonov regularizationwith ε = 10minus4

highest levels on the left-hand sides The dynamic rangewas chosen to give enough headroom for amplifications(+10 dB) and allow 30 dB below the average radiation tovisualize the deviations for the given dynamic range De-pending on the type of data the frequency axis was cho-sen to show all valid spectral information of the data Allplots are normalized and thus no information about ab-solute level mismatches can be derived from the figuresHowever level attenuation is a simple process whereasthe fine structure of the HRTF is most meaningful so nor-malized figures help visualizing these small differences

Dummy head 1 The BEM-simulated and range extrap-olated HRTFs show a general good match cf Figure 4The inward range extrapolation results in stronger rip-ples cf Figure 4c which is caused by the amplificationof higher orders in the spherical wave spectrum In con-trast outward range extrapolation attenuates higher ordersand thus always results in smoother functions

As no significant portions of the data coverage are miss-ing regularization is not required Nevertheless a smallregularization paramater of ε = 10minus8 was set in order tobe consistent with the other simulations in this section

As mentioned above BEM simulations have been lim-ited to frequencies below 6 kHz To study range extrapo-

77

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 4 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 5 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF measured at 30 cm b HRTF measured at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

lation for higher frequencies measured HRTFs were usedThe results clearly show that inward extrapolation createsnoisy artifacts whereas outward extrapolation results innoticeably smoother functions In general the data matchis good even if clearly visible deviations exist These devi-ations are most-likely caused by measurement uncertain-ties

Dummy head 2 The BEM simulation results of dummyhead 2 are depicted in Figure 6 and show a high matchof original and extrapolated functions even in most partsof the fine structure Some regions however show minordeviations inward extrapolation results in amplitude rip-ples and outward extrapolation yields small deviations forhigh frequencies This might be put down to the intersec-

78

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

(c) (d)

Figure 6 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 7 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulated at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

tions of the sphere of 30 cm radius with the torso of thedummy head 2 While the intersecting parts are removedsecondary sources close to the sphere still exist (caused byedge diffraction) which might create some errors A dis-continuity is clearly visible at 10 kHz in Figure 6 It orig-inates from a change of mesh size for BEM simulationsfor this dummy head as mentioned in section 31 As both

range extrapolation and simulation are performed for dis-crete frequency values the observed artifacts do not affectthe results negatively

Finally in Figure 7 measurements for dummy head 2are extrapolated in both directions and the results showa very good match between measured and extrapolatedHRTFs Here ripples not only occur for inward extrap-

79

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

The spatial resolution of measured HRTF data is gener-ally lower than the minimum audible angle of human audi-tory perception cf [1] To provide the listener with a highspatial resolution interpolation techniques have to be ap-plied Angular interpolation methods can be categorizedinto local interpolation only using neighboring HRTFs[14 15] and global interpolation based on the analysisand decomposition of the entire set of measured HRTFsKistler et al [16] and Chen et al [17] proposed the useof principal component analysis (PCA) for angular inter-polation Evans et al [18] first applied the spherical har-monics transform using a Gaussian quadrature method todetermine the optimal measurement points on the sphereThe latter method was further extended to irregular andopen spatial sampling grids using appropriate regulariza-tion methods [19] It was shown that the re-synthesizedHRTFs are in good agreement with the measured data

Human auditory distance perception is less accuratethan the ability to determine the angular direction of asound The perceived distance of a sound source in aroom mainly depends on the direct-sound-to-reverb ra-tio and the distance-dependent level attenuation and high-frequency damping These far-field effects are commonlyapplied to create distance perception in virtual audio en-vironments [20] In the proximity of the head auditorydistance perception appears to be determined by differentfactors Brungart and Rabinowitz [21] studied the phys-ical and perceptual differences of near-field and far-fieldHRTFs They have shown that the HRTF changes sig-nificantly for sources at distances less than 1 m from thehead More recently Lentz et al [22] perceptually evalu-ated measured HRTFs at different distances from the headshowing limits of noticeable differences between near-field and far-field HRTFs Borszlig [23] used a computation-ally efficient analytical model for the simulation of near-field sources which takes both the head-related and source-related near-field effects into account

In the proximal region HRTFs have to be measuredon a dense angular mesh at different distances from thehead which requires a significant number of measurementpoints Thus a crucial question is whether this data canbe calculated from far-field measurements at a single dis-tance Romblom and Cook [24] proposed near-field com-pensation filters and a geometric HRTF selection approachto simulate nearby sources Duraiswami et al [25] andZhang et al [26] extended the work of Evans et al [18] byfurther applying the radial wave propagation terms trans-forming the range extrapolation of HRTFs into an acousticradiation problem They evaluated simulation results on aspherical head model [27]

This article is organized as followed Section 2 reviewsthe spherical wave spectrum and range extrapolation inthe spherical harmonic domain It further introduces thenormalized correlation of spherical functions as a qualitymeasure for the simulation results HRTFs are typicallyderived from measurements over an incomplete portionof the sphere as the source directions in the lower hemi-sphere intersect with the subjectrsquos body Several strategies

are possible to obtain the spherical wave spectrum fromthese incomplete data sets The problems of matrix inver-sion and regularization are briefly discussed in this sectionIn section 3 range extrapolated HRTFs for two differentdummy heads are compared to measurement data in theproximal region and boundary-element simulation (BEM)results Section 4 analyzes the results and further comparesthe range extrapolation in the spherical wave spectral do-main with other range extrapolation methods proposed inliterature It is shown that the spherical acoustic methodprovides one efficient means of near-field binaural synthe-sis with a good agreement with measurement data

2 Interpolation and range extrapolation ofHRTFs

Applying the principle of reciprocity HRTFs can be re-formulated as an acoustic radiation problem [28] Whenthe volume velocity at the entrance of the ear canal isknown the radiated pressure field is completely definedoutside a region that encloses all contributing scatteringsources It can be calculated from the Helmholtz equationassuming that the Sommerfeld radiation condition is ful-filled [25 28]

21 Spherical wave spectrum

The solution of the wave equation in spherical coordinatesunder the assumption that all inward travelling waves arenegligible results in the expansion of the acoustic pressurefield

p(r θ φ k) =+infin

n=0

n

m=minusnanm(r k)Y m

n (θ φ) (1)

at position (r θ φ) and wavenumber k = 2πfc Hereby

Y mn (θ φ) = (minus1)m

(2n + 1)4π

(n minus |m|)(n + |m|)

middot P |m|n (cos θ)eimφ (2)

denotes the complex spherical harmonic function of ordern and degree m and

anm(r k) = bnm(k)hn(kr) (3)

the spherical expansion coefficients hn is the sphericalHankel function of the first kind which is associated withthe outgoing component of the sound field and P

|m|n are

the associated Legendre functions bnm denotes the spher-ical expansion coefficients which have to be determinedby a projection-based analysis

One can note from equations (1) and (3) that for a givenwave number k the sound pressure field is entirely de-termined by the spherical harmonic expansion coefficientsbnm(k) The prediction of HRTFs is thus realized in twoprocessing steps (a) the expansion coefficients anm(r0 k)are determined by expanding the set of measured HRTFs

73

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

into spherical harmonics on a sphere with radius r0 and(b) the HRTF at a target radius r1 is calculated by rangeextrapolation from radius r0 applying the correspondingspherical Hankel functions

In practice the outer summation in equation (1) is lim-ited to orders n le N which corresponds to a spatial bandlimitation It can be further written in matrix form as

p = Hb (4)

with p denoting the sound pressure vector of Np measure-ment points on a spherical shell and H the Np times (N + 1)2

matrix of spherical harmonics multiplied with the corre-sponding Hankel functions both limited to a maximumorder of N

The measured data at a known distance r0 allows toevaluate the HRTFs at any angular position by means ofthe spherical harmonics expansion with the coefficientsa(r0 k) given in equation (3)

The reproduction accuracy is limited by the order trun-cation as the spatial band limitation yields spatial alias-ing The range extrapolation problem can be solved by ap-plying the corresponding radial propagation functions tocompute the spherical harmonic coefficients anm(r1 k) attarget distance r1

anm(r1 k) = anm(r0 k)hn(kr1)hn(kr0)

(5)

and inserting these coefficients in equation (1) In order toavoid artifacts due to the exponential growth of the spher-ical Hankel functions for higher orders and small argu-ments kr the truncation order is chosen as N = krmin

with rmin being the radius of the smallest sphere enclosingthe listenerrsquos head with all its significant scattering sources[25]

22 Matrix inversion

The spherical harmonics decomposition which solvesequation (4) for the spherical harmonics coefficients b re-quires the inversion of matrix H Depending on the distri-bution of measurement points on the sphere this inversionproblem can be ill-posed ie the solution lacks stabilitywith respect to data perturbations Zotkin et al [19] deter-mined the most appropriate solving methods for differentsampling grids Gaussian irregularly sampled open andclosed grids have been studied

In this study non-Gaussian open grids (with a polargap) were used for most of the HRTF measurements Asa consequence the inversion must incorporate some sortof regularization In general regularization approximatesa solution and balances the approximation error versus thedata error for instance the solution becomes less sensitiveto data perturbations than the least-squares method [29]

A least-squares method with Tikhonov regularizationwas applied to calculate the expansion coefficients fromequation (4) as

breg = HHWH + εD minus1 HHWp (6)

where W = diagw is the Np times Np diagonal matrixof weighting coefficients and D the applied regularizationmatrix The weighting coefficients were obtained from theVoronoi surfaces associated with each measurement point[30] the regularization coefficient ε controls the amountof regularization and is usually set to small values [25]

Standard Tikhonov regularization Using standard Tik-honov regularization the regularization matrix D is chosento be the identity matrix of size (N + 1)2 times (N + 1)2

D = I (7)

Decomposition-order dependent Tikhonov regularizationDuraiswami et al [25] proposed a decomposition-orderdependent diagonal regularization matrix D

D = (1 + n(n + 1))I (8)

where I denotes the identity matrix and n the order of thespherical harmonics of the columns of matrix H

23 Correlation of spherical functions

In order to analyze the performance of the applied rangeextrapolation methods the correlation of two sphericalfunctions f and g can be used as a measure for their simi-larity It is defined as

C(f g) =S2

f (θ φ) g(θ φ) dΩ (9)

with the overbar denoting the complex conjugate cf [31]By transforming f and g into the spherical harmonic do-main the orthogonality of the spherical harmonics can beexploited Equation (9) thus writes as

C(f g) =infin

n=0

n

m=minusnfnmgnm (10)

The normalized correlation in the spherical harmonicdomain C(f g) which is defined as

C(f g) =C(f g)

EfEg

(11)

provides a good measure for the similarity of the sphericalshape of two functions It thus allows to compare spheri-cal functions independent of a possible gain mismatch Ef

and Eg denote the energies of the spherical functions fand g respectively

Using vector notation f = vecfnm the normalized cor-relation can be further expressed as

C(f g) =fHg

|f| middot |g| (12)

with || = 2 being the 2-norm of the coefficient vectorscf [32]

74

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

Figure 1 Geometries of the dummy heads used for simulationsand measurements a Dummy head 1 b Dummy head 2

3 HRTF measurement and numerical sim-ulation

For this study two different dummy heads which are de-picted in Figure 1 were used for both simulations andmeasurements The first herein referred to as ldquodummyhead 1rdquo is a HEAD acoustics HSU III mannequin thatprovides a relatively simple head torso and ear geome-try it is thus particularly suited for numerical simulationsThe second herein referred to as ldquodummy head 2rdquo is acustom-made mannequin produced at ITA Aachen with asimple torso but a much more detailed ear geometry [33]

31 Boundary-element method simulation

In order to obtain noise-free data for further comparisonsboundary-element-method (BEM) simulations were usedto calculate the HRTFs for the two dummy heads Thephysical structure of the heads was modeled using lin-ear triangular elements and the reciprocal acoustic methodwas applied for calculating the sound field around thehead propagation through the head structure was ignoredThe boundary condition can be defined by applying a con-stant surface velocity to the elements at the entrance ofthe ear canal on one side of the head With this boundarycondition the acoustic radiation was simulated to differ-ent field point grids at various distances from the centerof the head To calculate the directional transfer functions(DTFs) from the HRTFs the results of the BEM simulationwere referenced to the pressure created by a point sourceof equal volume velocity at the same radial distance TheDTFs are thus equal to the parts of the HRTFs whichvary for different directions of sound incidence while anyglobal frequency dependency for all directions of soundincidence is eliminated [16]

The LMS VirtualLab 9-SL3 software was used for BEMsimulations for this study the indirect frequency-domainBEM formulation was applied to all simulations

Dummy head 1 HRTFs for the first dummy head werecalculated for frequencies from 20 Hz to 6 kHz at 20 Hzsteps The field point grids are equiangular with an angu-lar resolution of 5 for both elevation and azimuth anglewith distances of 30 50 100 and 200 cm from the headThe simulation of the closest range contains some evalua-tion points that lie within the structure of the dummy head

and thus do not yield valid simulation results These pointswere omitted in the analysis

Dummy head 2 HRTFs for the second dummy headwere calculated for a frequency range from 20 Hz to 5 kHzwith a linear resolution of 20 Hz and from 5 kHz to themaximum frequency of 16 kHz at 50 Hz steps The sizeof the boundary elements directly relates to the upper fre-quency bound for which the calculation is valid A densecomputational mesh results in large BEM matrices andslows down the calculation speed Two meshes with differ-ent spatial discretization were applied to the simulationsthe first a coarse mesh for frequencies up to 10 kHz andthe second a fine mesh for frequencies above this limit

The field mesh was generated with a Gaussian samplingscheme of order 70 for the same distances from the headas for dummy head 1 This results in a high spatial reso-lution of approximately 2 in both elevation and azimuthThe calculation of the spherical wave spectrum was lim-ited to the order 35 and could thus be computed on a sin-gle workstation At the closest range of 30 cm some of themesh points lie inside the dummy head and are thus omit-ted from further calculations

32 Near-field HRTF measurement

For further comparisons HRTFs for both dummy headswere measured on a dense angular grid at several dis-tances from the head The different measurement setupsand methods are described in the following sections

Dummy head 1 ndash Outward (reciprocal) measurementThe HRTF data for dummy head 1 was measured inan anechoic chamber (59 m times 44 m times 425 m) at IR-CAM This anechoic room has a lower frequency boundof 75 Hz The measurement system consists of a Bruel ampKjaer turntable and a custom-built remote-controlled ro-tating arm that can be equipped with loudspeakers andormicrophones

The dummy head was positioned in the horizontal planeusing two laser beams for aligning the interaural axis withthe measurement device For HRTF measurements theacoustic reciprocity method [34] was applied The mea-surement signal was emitted from a Knowles ED29689miniature loudspeaker positioned at the entrance of the earcanal and recorded by a microphone array Eight MonacorMCE 2000 omnidirectional microphones were mountedon a 170 cm rod perpendicular to the mechanical armpointing towards the center of the dummy head The dis-tances between the microphones and the ipsilateral ear onthe ear axis were set to 20 30 50 70 100 136 170 and200 cm All microphones were powered and amplified bya STUDER LMS Carouso Mic24 ADAT preamplifier andsignals were played back and recorded with a RME Digi-face HDSP Hammerfall audio card with 48 kHz samplingrate The excitation signal was an exponential sweep [35]with a length of N = 216 samples A total number of 865impulse responses was measured for each radial distanceproviding sampling points from -30 to 90 elevation and

75

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

360 azimuth with a step size of 10 in elevation and 5

in azimuthAll microphones had been equalized to compensate for

temporal phase and level differences The closest micro-phone position at full lateralization ie on the interauralaxis was defined as global time reference point Diffuse-field equalization filters as described in [36 37] were ap-plied to all measurements These filters are derived froma weighted average over all measurements for one micro-phone at a known distance The applied weighting coeffi-cients correspond to the Voronoi surfaces [30] associatedwith each measurement position After the diffuse-fieldequalization directional transfer functions (DTFs) wereobtained for further calculations

Dummy head 2 ndash Inward measurement The near-fieldHRTFs for dummy head 2 were measured by Lentz [38]in a semi-anechoic chamber (11 m times 6 m times 5 m) at ITAAachen This anechoic room provides a lower cut-off fre-quency of approx 100 Hz Similarly to the setup used atIRCAM a computerized positioning system consisting ofa turntable and a rotary arm for a moving sound sourcewere used that allows to measure HRTFs in the proximalregion of the head

The dummy head was mounted and aligned on theturntable and a pair of microphones was positioned at theentrance of the ear canals using custom made earmoldshells The entire measurement equipment was calibratedand referenced to the measured sound pressure of a mi-crophone in absence of the dummy head yielding the di-rectional transfer functions (DTFs) Due to the solid floorin the semi-anechoic room a strong first ground reflectionoccurred which was eliminated by extracting the first partof the measured impulse response

An equiangular spatial sampling grid with an angularresolution of 5 in elevation and azimuthal direction wasapplied which results in 2664 measurement positions onthe full sphere In practice for source distances close tothe head only partial sphere measurements could be ob-tained Thus the coverage of the HRTF data varies withthe measurement distance Whereas for distances at 20 and30 cm only the upper hemispherical HRTF data could beobtained the measurements at distances 40 50 75 100and 200 cm cover larger parts of the sphere encompassingall elevations from +90 (above the head) to -45 -55-75 -75 and -90 respectively

A more detailed description of the measurement setupcan be found in [38]

4 Analysis of the results

In this section the results of the proposed methods aregiven First the correlation analysis is used to show thematch of original and extrapolated HRTFs as functions offrequency This provides the possibility to study the ef-fect of regularization on partially defined data Secondthe HRTFs in the horizontal plane are plotted over fre-quency which allows to observe the fine structure of origi-nal and calculated functions in the spatial domain Finally

1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 2 Correlation between measured HRTFs and extrapo-lation results from measurement data obtained at a differentrange (dummy head 1) a Calculation without regularization(weighted least squares) b Calculation using decomposition-order dependent regularization with ε = 10minus8

the given calculation using the spherical wave spectrum iscompared to alternative near-field compensation methodsfound in literature

41 Correlation analysis of the HRTF extrapolation

The normalized correlation function C(f g) as defined inequation (12) is used to evaluate the range extrapolationof HRTFs in the spherical wave spectral domain Therebythe lower bound C = 0 and the upper bound C = 1 cor-respond to a total mismatch and a perfect match of thesimulated and extrapolated data respectively

Measurement data The correlation over frequency formeasurement data of dummy head 1 is depicted in Fig-ure 2 As mentioned earlier diffuse field normalizationwas applied to all data sets before correlation the resultsthus refer to the directional transfer functions (DTFs) Asa result of missing measurement data around the southpole ie the polar gap the weighted least squares ap-proach fails To obtain accurate range extrapolation data inboth outward and inward directions regularization meth-ods have been applied The regularization matrix is givenin equation (8) and the regularization parameter was set toε = 10minus8 In doing so the high frequency part of the ana-lyzed HRTFs could be more accurately reproduced thanwhen using ε = 10minus6 as has been suggested in [25]As already mentioned in Sec 21 the maximum sphericalharmonic order was truncated to N = krmin see also[25] rmin was set to 30 cm and thus encloses all scatteringsources

The correlation measure shows similar results for boththe inward and the outward extrapolation of HRTFs Therelatively high correlation does not reflect differences inthe fine structure of the HRTFs and their dependencies onthe direction of range extrapolation This will be discussedin more detail in Sec 42 The correlation curves for bothinward and outward extrapolation follow the same trend

76

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

and show a drop above 6 kHz which might be caused bymeasurement uncertainties

BEM simulation data Numerical simulations allow toanalyze the HRTF range extrapolation and interpolationmethods without the effects of measurement noise For thisreason the BEM method described in Sec 31 was appliedto dummy head 1 The simulation results were truncatedto elevations from+90 (north pole) to -30 and thus coverthe same portion of the sphere as the measured HRTFs

Figure 3 depicts the correlation between range extrapo-lated and BEM simulated HRTFs derived from complete(full sphere) and incomplete (partial sphere) data sets andfor different inversion methods The range extrapolatedHRTFs are in a good agreement with the BEM simulationresults and show in general slightly higher accuracy fornear-field to far-field range extrapolation than vice versa

It can be seen that the correlation monotonically de-creases with increasing frequencies up to 180 Hz This isbecause the spherical harmonic of order zero (monopole)fails to model the spatial variations of the HRTF dataAbove this frequency the maximum order is set to one(monopole plus dipole) and the extrapolated HRTFs areagain well correlated with the BEM simulations Thiseffect can be minimized by introducing smooth (order-weighted) transitions or by increasing the maximum orderin this frequency region However due to the high corre-lation values (above 095) this has not been investigatedfurther in this study as it is unlikely to cause significantaudible artifacts

The correlation curves in Figures 3b and c illustrate theimportance of regularization for data from measurementsover an incomplete portion of the sphere Figure 3b clearlyshows that weighted least squares inversion fails for thepartial sphere at frequencies above 15 kHz with the cor-relation decreasing rapidly for higher frequencies In Fig-ure 3c Tikhonov regularization with a regularization pa-rameter ε = 10minus4 was applied and the extrapolated HRTFsare well correlated with the BEM simulated data over theentire frequency range It is important to note that the BEMsimulations were limited to frequencies up to 6 kHz How-ever measurement results for the same dummy head asdepicted in Figure 2 show a good agreement of range ex-trapolated and measured HRTFs over almost the entire fre-quency range of human hearing (up to 16 kHz) when reg-ularization is applied

42 Azimuthal frequency plots for outward and in-ward range extrapolation

In this section the results of inward and outward range ex-trapolation are compared graphically Only the azimuthalvariation (horizontal plane of the HRTFs) is depicted asa magnitude plot over frequency Hereby the radiation isnormalized to the average (omnidirectional) radiation foreach frequency each plot thus denotes the deviation fromthe average as level values in decibels

For all dummy heads the results for the left ear were an-alyzed each of the plots in Figures 4 to 8 thus show their

20 40 60 100 200 400 1k 2k 4k 6k09

095

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

(c)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 3 Correlation between extrapolated HRTFs and corre-sponding BEM simulation results for dummy head 1 at dis-tances 200 cm (solid line) and 30 cm (dashed line) The range ex-trapolation is calculated from BEM simulated HRTF data whichcovers (a) the full sphere or (b)ndash(c) only a partial sphere witha polar gap below minus30 elevation (with and without regulariza-tion) Thus ldquo30 cmrarr200 cm denotes range extrapolation fromradius 30 cm to 200 cm compared to BEM simulation results atradius 200 cm a Full sphere without regularization (weightedleast squares) b Partial sphere without regularization (weightedleast squares) c Partial sphere standard Tikhonov regularizationwith ε = 10minus4

highest levels on the left-hand sides The dynamic rangewas chosen to give enough headroom for amplifications(+10 dB) and allow 30 dB below the average radiation tovisualize the deviations for the given dynamic range De-pending on the type of data the frequency axis was cho-sen to show all valid spectral information of the data Allplots are normalized and thus no information about ab-solute level mismatches can be derived from the figuresHowever level attenuation is a simple process whereasthe fine structure of the HRTF is most meaningful so nor-malized figures help visualizing these small differences

Dummy head 1 The BEM-simulated and range extrap-olated HRTFs show a general good match cf Figure 4The inward range extrapolation results in stronger rip-ples cf Figure 4c which is caused by the amplificationof higher orders in the spherical wave spectrum In con-trast outward range extrapolation attenuates higher ordersand thus always results in smoother functions

As no significant portions of the data coverage are miss-ing regularization is not required Nevertheless a smallregularization paramater of ε = 10minus8 was set in order tobe consistent with the other simulations in this section

As mentioned above BEM simulations have been lim-ited to frequencies below 6 kHz To study range extrapo-

77

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 4 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 5 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF measured at 30 cm b HRTF measured at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

lation for higher frequencies measured HRTFs were usedThe results clearly show that inward extrapolation createsnoisy artifacts whereas outward extrapolation results innoticeably smoother functions In general the data matchis good even if clearly visible deviations exist These devi-ations are most-likely caused by measurement uncertain-ties

Dummy head 2 The BEM simulation results of dummyhead 2 are depicted in Figure 6 and show a high matchof original and extrapolated functions even in most partsof the fine structure Some regions however show minordeviations inward extrapolation results in amplitude rip-ples and outward extrapolation yields small deviations forhigh frequencies This might be put down to the intersec-

78

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

(c) (d)

Figure 6 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 7 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulated at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

tions of the sphere of 30 cm radius with the torso of thedummy head 2 While the intersecting parts are removedsecondary sources close to the sphere still exist (caused byedge diffraction) which might create some errors A dis-continuity is clearly visible at 10 kHz in Figure 6 It orig-inates from a change of mesh size for BEM simulationsfor this dummy head as mentioned in section 31 As both

range extrapolation and simulation are performed for dis-crete frequency values the observed artifacts do not affectthe results negatively

Finally in Figure 7 measurements for dummy head 2are extrapolated in both directions and the results showa very good match between measured and extrapolatedHRTFs Here ripples not only occur for inward extrap-

79

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

into spherical harmonics on a sphere with radius r0 and(b) the HRTF at a target radius r1 is calculated by rangeextrapolation from radius r0 applying the correspondingspherical Hankel functions

In practice the outer summation in equation (1) is lim-ited to orders n le N which corresponds to a spatial bandlimitation It can be further written in matrix form as

p = Hb (4)

with p denoting the sound pressure vector of Np measure-ment points on a spherical shell and H the Np times (N + 1)2

matrix of spherical harmonics multiplied with the corre-sponding Hankel functions both limited to a maximumorder of N

The measured data at a known distance r0 allows toevaluate the HRTFs at any angular position by means ofthe spherical harmonics expansion with the coefficientsa(r0 k) given in equation (3)

The reproduction accuracy is limited by the order trun-cation as the spatial band limitation yields spatial alias-ing The range extrapolation problem can be solved by ap-plying the corresponding radial propagation functions tocompute the spherical harmonic coefficients anm(r1 k) attarget distance r1

anm(r1 k) = anm(r0 k)hn(kr1)hn(kr0)

(5)

and inserting these coefficients in equation (1) In order toavoid artifacts due to the exponential growth of the spher-ical Hankel functions for higher orders and small argu-ments kr the truncation order is chosen as N = krmin

with rmin being the radius of the smallest sphere enclosingthe listenerrsquos head with all its significant scattering sources[25]

22 Matrix inversion

The spherical harmonics decomposition which solvesequation (4) for the spherical harmonics coefficients b re-quires the inversion of matrix H Depending on the distri-bution of measurement points on the sphere this inversionproblem can be ill-posed ie the solution lacks stabilitywith respect to data perturbations Zotkin et al [19] deter-mined the most appropriate solving methods for differentsampling grids Gaussian irregularly sampled open andclosed grids have been studied

In this study non-Gaussian open grids (with a polargap) were used for most of the HRTF measurements Asa consequence the inversion must incorporate some sortof regularization In general regularization approximatesa solution and balances the approximation error versus thedata error for instance the solution becomes less sensitiveto data perturbations than the least-squares method [29]

A least-squares method with Tikhonov regularizationwas applied to calculate the expansion coefficients fromequation (4) as

breg = HHWH + εD minus1 HHWp (6)

where W = diagw is the Np times Np diagonal matrixof weighting coefficients and D the applied regularizationmatrix The weighting coefficients were obtained from theVoronoi surfaces associated with each measurement point[30] the regularization coefficient ε controls the amountof regularization and is usually set to small values [25]

Standard Tikhonov regularization Using standard Tik-honov regularization the regularization matrix D is chosento be the identity matrix of size (N + 1)2 times (N + 1)2

D = I (7)

Decomposition-order dependent Tikhonov regularizationDuraiswami et al [25] proposed a decomposition-orderdependent diagonal regularization matrix D

D = (1 + n(n + 1))I (8)

where I denotes the identity matrix and n the order of thespherical harmonics of the columns of matrix H

23 Correlation of spherical functions

In order to analyze the performance of the applied rangeextrapolation methods the correlation of two sphericalfunctions f and g can be used as a measure for their simi-larity It is defined as

C(f g) =S2

f (θ φ) g(θ φ) dΩ (9)

with the overbar denoting the complex conjugate cf [31]By transforming f and g into the spherical harmonic do-main the orthogonality of the spherical harmonics can beexploited Equation (9) thus writes as

C(f g) =infin

n=0

n

m=minusnfnmgnm (10)

The normalized correlation in the spherical harmonicdomain C(f g) which is defined as

C(f g) =C(f g)

EfEg

(11)

provides a good measure for the similarity of the sphericalshape of two functions It thus allows to compare spheri-cal functions independent of a possible gain mismatch Ef

and Eg denote the energies of the spherical functions fand g respectively

Using vector notation f = vecfnm the normalized cor-relation can be further expressed as

C(f g) =fHg

|f| middot |g| (12)

with || = 2 being the 2-norm of the coefficient vectorscf [32]

74

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

Figure 1 Geometries of the dummy heads used for simulationsand measurements a Dummy head 1 b Dummy head 2

3 HRTF measurement and numerical sim-ulation

For this study two different dummy heads which are de-picted in Figure 1 were used for both simulations andmeasurements The first herein referred to as ldquodummyhead 1rdquo is a HEAD acoustics HSU III mannequin thatprovides a relatively simple head torso and ear geome-try it is thus particularly suited for numerical simulationsThe second herein referred to as ldquodummy head 2rdquo is acustom-made mannequin produced at ITA Aachen with asimple torso but a much more detailed ear geometry [33]

31 Boundary-element method simulation

In order to obtain noise-free data for further comparisonsboundary-element-method (BEM) simulations were usedto calculate the HRTFs for the two dummy heads Thephysical structure of the heads was modeled using lin-ear triangular elements and the reciprocal acoustic methodwas applied for calculating the sound field around thehead propagation through the head structure was ignoredThe boundary condition can be defined by applying a con-stant surface velocity to the elements at the entrance ofthe ear canal on one side of the head With this boundarycondition the acoustic radiation was simulated to differ-ent field point grids at various distances from the centerof the head To calculate the directional transfer functions(DTFs) from the HRTFs the results of the BEM simulationwere referenced to the pressure created by a point sourceof equal volume velocity at the same radial distance TheDTFs are thus equal to the parts of the HRTFs whichvary for different directions of sound incidence while anyglobal frequency dependency for all directions of soundincidence is eliminated [16]

The LMS VirtualLab 9-SL3 software was used for BEMsimulations for this study the indirect frequency-domainBEM formulation was applied to all simulations

Dummy head 1 HRTFs for the first dummy head werecalculated for frequencies from 20 Hz to 6 kHz at 20 Hzsteps The field point grids are equiangular with an angu-lar resolution of 5 for both elevation and azimuth anglewith distances of 30 50 100 and 200 cm from the headThe simulation of the closest range contains some evalua-tion points that lie within the structure of the dummy head

and thus do not yield valid simulation results These pointswere omitted in the analysis

Dummy head 2 HRTFs for the second dummy headwere calculated for a frequency range from 20 Hz to 5 kHzwith a linear resolution of 20 Hz and from 5 kHz to themaximum frequency of 16 kHz at 50 Hz steps The sizeof the boundary elements directly relates to the upper fre-quency bound for which the calculation is valid A densecomputational mesh results in large BEM matrices andslows down the calculation speed Two meshes with differ-ent spatial discretization were applied to the simulationsthe first a coarse mesh for frequencies up to 10 kHz andthe second a fine mesh for frequencies above this limit

The field mesh was generated with a Gaussian samplingscheme of order 70 for the same distances from the headas for dummy head 1 This results in a high spatial reso-lution of approximately 2 in both elevation and azimuthThe calculation of the spherical wave spectrum was lim-ited to the order 35 and could thus be computed on a sin-gle workstation At the closest range of 30 cm some of themesh points lie inside the dummy head and are thus omit-ted from further calculations

32 Near-field HRTF measurement

For further comparisons HRTFs for both dummy headswere measured on a dense angular grid at several dis-tances from the head The different measurement setupsand methods are described in the following sections

Dummy head 1 ndash Outward (reciprocal) measurementThe HRTF data for dummy head 1 was measured inan anechoic chamber (59 m times 44 m times 425 m) at IR-CAM This anechoic room has a lower frequency boundof 75 Hz The measurement system consists of a Bruel ampKjaer turntable and a custom-built remote-controlled ro-tating arm that can be equipped with loudspeakers andormicrophones

The dummy head was positioned in the horizontal planeusing two laser beams for aligning the interaural axis withthe measurement device For HRTF measurements theacoustic reciprocity method [34] was applied The mea-surement signal was emitted from a Knowles ED29689miniature loudspeaker positioned at the entrance of the earcanal and recorded by a microphone array Eight MonacorMCE 2000 omnidirectional microphones were mountedon a 170 cm rod perpendicular to the mechanical armpointing towards the center of the dummy head The dis-tances between the microphones and the ipsilateral ear onthe ear axis were set to 20 30 50 70 100 136 170 and200 cm All microphones were powered and amplified bya STUDER LMS Carouso Mic24 ADAT preamplifier andsignals were played back and recorded with a RME Digi-face HDSP Hammerfall audio card with 48 kHz samplingrate The excitation signal was an exponential sweep [35]with a length of N = 216 samples A total number of 865impulse responses was measured for each radial distanceproviding sampling points from -30 to 90 elevation and

75

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

360 azimuth with a step size of 10 in elevation and 5

in azimuthAll microphones had been equalized to compensate for

temporal phase and level differences The closest micro-phone position at full lateralization ie on the interauralaxis was defined as global time reference point Diffuse-field equalization filters as described in [36 37] were ap-plied to all measurements These filters are derived froma weighted average over all measurements for one micro-phone at a known distance The applied weighting coeffi-cients correspond to the Voronoi surfaces [30] associatedwith each measurement position After the diffuse-fieldequalization directional transfer functions (DTFs) wereobtained for further calculations

Dummy head 2 ndash Inward measurement The near-fieldHRTFs for dummy head 2 were measured by Lentz [38]in a semi-anechoic chamber (11 m times 6 m times 5 m) at ITAAachen This anechoic room provides a lower cut-off fre-quency of approx 100 Hz Similarly to the setup used atIRCAM a computerized positioning system consisting ofa turntable and a rotary arm for a moving sound sourcewere used that allows to measure HRTFs in the proximalregion of the head

The dummy head was mounted and aligned on theturntable and a pair of microphones was positioned at theentrance of the ear canals using custom made earmoldshells The entire measurement equipment was calibratedand referenced to the measured sound pressure of a mi-crophone in absence of the dummy head yielding the di-rectional transfer functions (DTFs) Due to the solid floorin the semi-anechoic room a strong first ground reflectionoccurred which was eliminated by extracting the first partof the measured impulse response

An equiangular spatial sampling grid with an angularresolution of 5 in elevation and azimuthal direction wasapplied which results in 2664 measurement positions onthe full sphere In practice for source distances close tothe head only partial sphere measurements could be ob-tained Thus the coverage of the HRTF data varies withthe measurement distance Whereas for distances at 20 and30 cm only the upper hemispherical HRTF data could beobtained the measurements at distances 40 50 75 100and 200 cm cover larger parts of the sphere encompassingall elevations from +90 (above the head) to -45 -55-75 -75 and -90 respectively

A more detailed description of the measurement setupcan be found in [38]

4 Analysis of the results

In this section the results of the proposed methods aregiven First the correlation analysis is used to show thematch of original and extrapolated HRTFs as functions offrequency This provides the possibility to study the ef-fect of regularization on partially defined data Secondthe HRTFs in the horizontal plane are plotted over fre-quency which allows to observe the fine structure of origi-nal and calculated functions in the spatial domain Finally

1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 2 Correlation between measured HRTFs and extrapo-lation results from measurement data obtained at a differentrange (dummy head 1) a Calculation without regularization(weighted least squares) b Calculation using decomposition-order dependent regularization with ε = 10minus8

the given calculation using the spherical wave spectrum iscompared to alternative near-field compensation methodsfound in literature

41 Correlation analysis of the HRTF extrapolation

The normalized correlation function C(f g) as defined inequation (12) is used to evaluate the range extrapolationof HRTFs in the spherical wave spectral domain Therebythe lower bound C = 0 and the upper bound C = 1 cor-respond to a total mismatch and a perfect match of thesimulated and extrapolated data respectively

Measurement data The correlation over frequency formeasurement data of dummy head 1 is depicted in Fig-ure 2 As mentioned earlier diffuse field normalizationwas applied to all data sets before correlation the resultsthus refer to the directional transfer functions (DTFs) Asa result of missing measurement data around the southpole ie the polar gap the weighted least squares ap-proach fails To obtain accurate range extrapolation data inboth outward and inward directions regularization meth-ods have been applied The regularization matrix is givenin equation (8) and the regularization parameter was set toε = 10minus8 In doing so the high frequency part of the ana-lyzed HRTFs could be more accurately reproduced thanwhen using ε = 10minus6 as has been suggested in [25]As already mentioned in Sec 21 the maximum sphericalharmonic order was truncated to N = krmin see also[25] rmin was set to 30 cm and thus encloses all scatteringsources

The correlation measure shows similar results for boththe inward and the outward extrapolation of HRTFs Therelatively high correlation does not reflect differences inthe fine structure of the HRTFs and their dependencies onthe direction of range extrapolation This will be discussedin more detail in Sec 42 The correlation curves for bothinward and outward extrapolation follow the same trend

76

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

and show a drop above 6 kHz which might be caused bymeasurement uncertainties

BEM simulation data Numerical simulations allow toanalyze the HRTF range extrapolation and interpolationmethods without the effects of measurement noise For thisreason the BEM method described in Sec 31 was appliedto dummy head 1 The simulation results were truncatedto elevations from+90 (north pole) to -30 and thus coverthe same portion of the sphere as the measured HRTFs

Figure 3 depicts the correlation between range extrapo-lated and BEM simulated HRTFs derived from complete(full sphere) and incomplete (partial sphere) data sets andfor different inversion methods The range extrapolatedHRTFs are in a good agreement with the BEM simulationresults and show in general slightly higher accuracy fornear-field to far-field range extrapolation than vice versa

It can be seen that the correlation monotonically de-creases with increasing frequencies up to 180 Hz This isbecause the spherical harmonic of order zero (monopole)fails to model the spatial variations of the HRTF dataAbove this frequency the maximum order is set to one(monopole plus dipole) and the extrapolated HRTFs areagain well correlated with the BEM simulations Thiseffect can be minimized by introducing smooth (order-weighted) transitions or by increasing the maximum orderin this frequency region However due to the high corre-lation values (above 095) this has not been investigatedfurther in this study as it is unlikely to cause significantaudible artifacts

The correlation curves in Figures 3b and c illustrate theimportance of regularization for data from measurementsover an incomplete portion of the sphere Figure 3b clearlyshows that weighted least squares inversion fails for thepartial sphere at frequencies above 15 kHz with the cor-relation decreasing rapidly for higher frequencies In Fig-ure 3c Tikhonov regularization with a regularization pa-rameter ε = 10minus4 was applied and the extrapolated HRTFsare well correlated with the BEM simulated data over theentire frequency range It is important to note that the BEMsimulations were limited to frequencies up to 6 kHz How-ever measurement results for the same dummy head asdepicted in Figure 2 show a good agreement of range ex-trapolated and measured HRTFs over almost the entire fre-quency range of human hearing (up to 16 kHz) when reg-ularization is applied

42 Azimuthal frequency plots for outward and in-ward range extrapolation

In this section the results of inward and outward range ex-trapolation are compared graphically Only the azimuthalvariation (horizontal plane of the HRTFs) is depicted asa magnitude plot over frequency Hereby the radiation isnormalized to the average (omnidirectional) radiation foreach frequency each plot thus denotes the deviation fromthe average as level values in decibels

For all dummy heads the results for the left ear were an-alyzed each of the plots in Figures 4 to 8 thus show their

20 40 60 100 200 400 1k 2k 4k 6k09

095

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

(c)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 3 Correlation between extrapolated HRTFs and corre-sponding BEM simulation results for dummy head 1 at dis-tances 200 cm (solid line) and 30 cm (dashed line) The range ex-trapolation is calculated from BEM simulated HRTF data whichcovers (a) the full sphere or (b)ndash(c) only a partial sphere witha polar gap below minus30 elevation (with and without regulariza-tion) Thus ldquo30 cmrarr200 cm denotes range extrapolation fromradius 30 cm to 200 cm compared to BEM simulation results atradius 200 cm a Full sphere without regularization (weightedleast squares) b Partial sphere without regularization (weightedleast squares) c Partial sphere standard Tikhonov regularizationwith ε = 10minus4

highest levels on the left-hand sides The dynamic rangewas chosen to give enough headroom for amplifications(+10 dB) and allow 30 dB below the average radiation tovisualize the deviations for the given dynamic range De-pending on the type of data the frequency axis was cho-sen to show all valid spectral information of the data Allplots are normalized and thus no information about ab-solute level mismatches can be derived from the figuresHowever level attenuation is a simple process whereasthe fine structure of the HRTF is most meaningful so nor-malized figures help visualizing these small differences

Dummy head 1 The BEM-simulated and range extrap-olated HRTFs show a general good match cf Figure 4The inward range extrapolation results in stronger rip-ples cf Figure 4c which is caused by the amplificationof higher orders in the spherical wave spectrum In con-trast outward range extrapolation attenuates higher ordersand thus always results in smoother functions

As no significant portions of the data coverage are miss-ing regularization is not required Nevertheless a smallregularization paramater of ε = 10minus8 was set in order tobe consistent with the other simulations in this section

As mentioned above BEM simulations have been lim-ited to frequencies below 6 kHz To study range extrapo-

77

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 4 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 5 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF measured at 30 cm b HRTF measured at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

lation for higher frequencies measured HRTFs were usedThe results clearly show that inward extrapolation createsnoisy artifacts whereas outward extrapolation results innoticeably smoother functions In general the data matchis good even if clearly visible deviations exist These devi-ations are most-likely caused by measurement uncertain-ties

Dummy head 2 The BEM simulation results of dummyhead 2 are depicted in Figure 6 and show a high matchof original and extrapolated functions even in most partsof the fine structure Some regions however show minordeviations inward extrapolation results in amplitude rip-ples and outward extrapolation yields small deviations forhigh frequencies This might be put down to the intersec-

78

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

(c) (d)

Figure 6 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 7 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulated at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

tions of the sphere of 30 cm radius with the torso of thedummy head 2 While the intersecting parts are removedsecondary sources close to the sphere still exist (caused byedge diffraction) which might create some errors A dis-continuity is clearly visible at 10 kHz in Figure 6 It orig-inates from a change of mesh size for BEM simulationsfor this dummy head as mentioned in section 31 As both

range extrapolation and simulation are performed for dis-crete frequency values the observed artifacts do not affectthe results negatively

Finally in Figure 7 measurements for dummy head 2are extrapolated in both directions and the results showa very good match between measured and extrapolatedHRTFs Here ripples not only occur for inward extrap-

79

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

Figure 1 Geometries of the dummy heads used for simulationsand measurements a Dummy head 1 b Dummy head 2

3 HRTF measurement and numerical sim-ulation

For this study two different dummy heads which are de-picted in Figure 1 were used for both simulations andmeasurements The first herein referred to as ldquodummyhead 1rdquo is a HEAD acoustics HSU III mannequin thatprovides a relatively simple head torso and ear geome-try it is thus particularly suited for numerical simulationsThe second herein referred to as ldquodummy head 2rdquo is acustom-made mannequin produced at ITA Aachen with asimple torso but a much more detailed ear geometry [33]

31 Boundary-element method simulation

In order to obtain noise-free data for further comparisonsboundary-element-method (BEM) simulations were usedto calculate the HRTFs for the two dummy heads Thephysical structure of the heads was modeled using lin-ear triangular elements and the reciprocal acoustic methodwas applied for calculating the sound field around thehead propagation through the head structure was ignoredThe boundary condition can be defined by applying a con-stant surface velocity to the elements at the entrance ofthe ear canal on one side of the head With this boundarycondition the acoustic radiation was simulated to differ-ent field point grids at various distances from the centerof the head To calculate the directional transfer functions(DTFs) from the HRTFs the results of the BEM simulationwere referenced to the pressure created by a point sourceof equal volume velocity at the same radial distance TheDTFs are thus equal to the parts of the HRTFs whichvary for different directions of sound incidence while anyglobal frequency dependency for all directions of soundincidence is eliminated [16]

The LMS VirtualLab 9-SL3 software was used for BEMsimulations for this study the indirect frequency-domainBEM formulation was applied to all simulations

Dummy head 1 HRTFs for the first dummy head werecalculated for frequencies from 20 Hz to 6 kHz at 20 Hzsteps The field point grids are equiangular with an angu-lar resolution of 5 for both elevation and azimuth anglewith distances of 30 50 100 and 200 cm from the headThe simulation of the closest range contains some evalua-tion points that lie within the structure of the dummy head

and thus do not yield valid simulation results These pointswere omitted in the analysis

Dummy head 2 HRTFs for the second dummy headwere calculated for a frequency range from 20 Hz to 5 kHzwith a linear resolution of 20 Hz and from 5 kHz to themaximum frequency of 16 kHz at 50 Hz steps The sizeof the boundary elements directly relates to the upper fre-quency bound for which the calculation is valid A densecomputational mesh results in large BEM matrices andslows down the calculation speed Two meshes with differ-ent spatial discretization were applied to the simulationsthe first a coarse mesh for frequencies up to 10 kHz andthe second a fine mesh for frequencies above this limit

The field mesh was generated with a Gaussian samplingscheme of order 70 for the same distances from the headas for dummy head 1 This results in a high spatial reso-lution of approximately 2 in both elevation and azimuthThe calculation of the spherical wave spectrum was lim-ited to the order 35 and could thus be computed on a sin-gle workstation At the closest range of 30 cm some of themesh points lie inside the dummy head and are thus omit-ted from further calculations

32 Near-field HRTF measurement

For further comparisons HRTFs for both dummy headswere measured on a dense angular grid at several dis-tances from the head The different measurement setupsand methods are described in the following sections

Dummy head 1 ndash Outward (reciprocal) measurementThe HRTF data for dummy head 1 was measured inan anechoic chamber (59 m times 44 m times 425 m) at IR-CAM This anechoic room has a lower frequency boundof 75 Hz The measurement system consists of a Bruel ampKjaer turntable and a custom-built remote-controlled ro-tating arm that can be equipped with loudspeakers andormicrophones

The dummy head was positioned in the horizontal planeusing two laser beams for aligning the interaural axis withthe measurement device For HRTF measurements theacoustic reciprocity method [34] was applied The mea-surement signal was emitted from a Knowles ED29689miniature loudspeaker positioned at the entrance of the earcanal and recorded by a microphone array Eight MonacorMCE 2000 omnidirectional microphones were mountedon a 170 cm rod perpendicular to the mechanical armpointing towards the center of the dummy head The dis-tances between the microphones and the ipsilateral ear onthe ear axis were set to 20 30 50 70 100 136 170 and200 cm All microphones were powered and amplified bya STUDER LMS Carouso Mic24 ADAT preamplifier andsignals were played back and recorded with a RME Digi-face HDSP Hammerfall audio card with 48 kHz samplingrate The excitation signal was an exponential sweep [35]with a length of N = 216 samples A total number of 865impulse responses was measured for each radial distanceproviding sampling points from -30 to 90 elevation and

75

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

360 azimuth with a step size of 10 in elevation and 5

in azimuthAll microphones had been equalized to compensate for

temporal phase and level differences The closest micro-phone position at full lateralization ie on the interauralaxis was defined as global time reference point Diffuse-field equalization filters as described in [36 37] were ap-plied to all measurements These filters are derived froma weighted average over all measurements for one micro-phone at a known distance The applied weighting coeffi-cients correspond to the Voronoi surfaces [30] associatedwith each measurement position After the diffuse-fieldequalization directional transfer functions (DTFs) wereobtained for further calculations

Dummy head 2 ndash Inward measurement The near-fieldHRTFs for dummy head 2 were measured by Lentz [38]in a semi-anechoic chamber (11 m times 6 m times 5 m) at ITAAachen This anechoic room provides a lower cut-off fre-quency of approx 100 Hz Similarly to the setup used atIRCAM a computerized positioning system consisting ofa turntable and a rotary arm for a moving sound sourcewere used that allows to measure HRTFs in the proximalregion of the head

The dummy head was mounted and aligned on theturntable and a pair of microphones was positioned at theentrance of the ear canals using custom made earmoldshells The entire measurement equipment was calibratedand referenced to the measured sound pressure of a mi-crophone in absence of the dummy head yielding the di-rectional transfer functions (DTFs) Due to the solid floorin the semi-anechoic room a strong first ground reflectionoccurred which was eliminated by extracting the first partof the measured impulse response

An equiangular spatial sampling grid with an angularresolution of 5 in elevation and azimuthal direction wasapplied which results in 2664 measurement positions onthe full sphere In practice for source distances close tothe head only partial sphere measurements could be ob-tained Thus the coverage of the HRTF data varies withthe measurement distance Whereas for distances at 20 and30 cm only the upper hemispherical HRTF data could beobtained the measurements at distances 40 50 75 100and 200 cm cover larger parts of the sphere encompassingall elevations from +90 (above the head) to -45 -55-75 -75 and -90 respectively

A more detailed description of the measurement setupcan be found in [38]

4 Analysis of the results

In this section the results of the proposed methods aregiven First the correlation analysis is used to show thematch of original and extrapolated HRTFs as functions offrequency This provides the possibility to study the ef-fect of regularization on partially defined data Secondthe HRTFs in the horizontal plane are plotted over fre-quency which allows to observe the fine structure of origi-nal and calculated functions in the spatial domain Finally

1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 2 Correlation between measured HRTFs and extrapo-lation results from measurement data obtained at a differentrange (dummy head 1) a Calculation without regularization(weighted least squares) b Calculation using decomposition-order dependent regularization with ε = 10minus8

the given calculation using the spherical wave spectrum iscompared to alternative near-field compensation methodsfound in literature

41 Correlation analysis of the HRTF extrapolation

The normalized correlation function C(f g) as defined inequation (12) is used to evaluate the range extrapolationof HRTFs in the spherical wave spectral domain Therebythe lower bound C = 0 and the upper bound C = 1 cor-respond to a total mismatch and a perfect match of thesimulated and extrapolated data respectively

Measurement data The correlation over frequency formeasurement data of dummy head 1 is depicted in Fig-ure 2 As mentioned earlier diffuse field normalizationwas applied to all data sets before correlation the resultsthus refer to the directional transfer functions (DTFs) Asa result of missing measurement data around the southpole ie the polar gap the weighted least squares ap-proach fails To obtain accurate range extrapolation data inboth outward and inward directions regularization meth-ods have been applied The regularization matrix is givenin equation (8) and the regularization parameter was set toε = 10minus8 In doing so the high frequency part of the ana-lyzed HRTFs could be more accurately reproduced thanwhen using ε = 10minus6 as has been suggested in [25]As already mentioned in Sec 21 the maximum sphericalharmonic order was truncated to N = krmin see also[25] rmin was set to 30 cm and thus encloses all scatteringsources

The correlation measure shows similar results for boththe inward and the outward extrapolation of HRTFs Therelatively high correlation does not reflect differences inthe fine structure of the HRTFs and their dependencies onthe direction of range extrapolation This will be discussedin more detail in Sec 42 The correlation curves for bothinward and outward extrapolation follow the same trend

76

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

and show a drop above 6 kHz which might be caused bymeasurement uncertainties

BEM simulation data Numerical simulations allow toanalyze the HRTF range extrapolation and interpolationmethods without the effects of measurement noise For thisreason the BEM method described in Sec 31 was appliedto dummy head 1 The simulation results were truncatedto elevations from+90 (north pole) to -30 and thus coverthe same portion of the sphere as the measured HRTFs

Figure 3 depicts the correlation between range extrapo-lated and BEM simulated HRTFs derived from complete(full sphere) and incomplete (partial sphere) data sets andfor different inversion methods The range extrapolatedHRTFs are in a good agreement with the BEM simulationresults and show in general slightly higher accuracy fornear-field to far-field range extrapolation than vice versa

It can be seen that the correlation monotonically de-creases with increasing frequencies up to 180 Hz This isbecause the spherical harmonic of order zero (monopole)fails to model the spatial variations of the HRTF dataAbove this frequency the maximum order is set to one(monopole plus dipole) and the extrapolated HRTFs areagain well correlated with the BEM simulations Thiseffect can be minimized by introducing smooth (order-weighted) transitions or by increasing the maximum orderin this frequency region However due to the high corre-lation values (above 095) this has not been investigatedfurther in this study as it is unlikely to cause significantaudible artifacts

The correlation curves in Figures 3b and c illustrate theimportance of regularization for data from measurementsover an incomplete portion of the sphere Figure 3b clearlyshows that weighted least squares inversion fails for thepartial sphere at frequencies above 15 kHz with the cor-relation decreasing rapidly for higher frequencies In Fig-ure 3c Tikhonov regularization with a regularization pa-rameter ε = 10minus4 was applied and the extrapolated HRTFsare well correlated with the BEM simulated data over theentire frequency range It is important to note that the BEMsimulations were limited to frequencies up to 6 kHz How-ever measurement results for the same dummy head asdepicted in Figure 2 show a good agreement of range ex-trapolated and measured HRTFs over almost the entire fre-quency range of human hearing (up to 16 kHz) when reg-ularization is applied

42 Azimuthal frequency plots for outward and in-ward range extrapolation

In this section the results of inward and outward range ex-trapolation are compared graphically Only the azimuthalvariation (horizontal plane of the HRTFs) is depicted asa magnitude plot over frequency Hereby the radiation isnormalized to the average (omnidirectional) radiation foreach frequency each plot thus denotes the deviation fromthe average as level values in decibels

For all dummy heads the results for the left ear were an-alyzed each of the plots in Figures 4 to 8 thus show their

20 40 60 100 200 400 1k 2k 4k 6k09

095

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

(c)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 3 Correlation between extrapolated HRTFs and corre-sponding BEM simulation results for dummy head 1 at dis-tances 200 cm (solid line) and 30 cm (dashed line) The range ex-trapolation is calculated from BEM simulated HRTF data whichcovers (a) the full sphere or (b)ndash(c) only a partial sphere witha polar gap below minus30 elevation (with and without regulariza-tion) Thus ldquo30 cmrarr200 cm denotes range extrapolation fromradius 30 cm to 200 cm compared to BEM simulation results atradius 200 cm a Full sphere without regularization (weightedleast squares) b Partial sphere without regularization (weightedleast squares) c Partial sphere standard Tikhonov regularizationwith ε = 10minus4

highest levels on the left-hand sides The dynamic rangewas chosen to give enough headroom for amplifications(+10 dB) and allow 30 dB below the average radiation tovisualize the deviations for the given dynamic range De-pending on the type of data the frequency axis was cho-sen to show all valid spectral information of the data Allplots are normalized and thus no information about ab-solute level mismatches can be derived from the figuresHowever level attenuation is a simple process whereasthe fine structure of the HRTF is most meaningful so nor-malized figures help visualizing these small differences

Dummy head 1 The BEM-simulated and range extrap-olated HRTFs show a general good match cf Figure 4The inward range extrapolation results in stronger rip-ples cf Figure 4c which is caused by the amplificationof higher orders in the spherical wave spectrum In con-trast outward range extrapolation attenuates higher ordersand thus always results in smoother functions

As no significant portions of the data coverage are miss-ing regularization is not required Nevertheless a smallregularization paramater of ε = 10minus8 was set in order tobe consistent with the other simulations in this section

As mentioned above BEM simulations have been lim-ited to frequencies below 6 kHz To study range extrapo-

77

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 4 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 5 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF measured at 30 cm b HRTF measured at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

lation for higher frequencies measured HRTFs were usedThe results clearly show that inward extrapolation createsnoisy artifacts whereas outward extrapolation results innoticeably smoother functions In general the data matchis good even if clearly visible deviations exist These devi-ations are most-likely caused by measurement uncertain-ties

Dummy head 2 The BEM simulation results of dummyhead 2 are depicted in Figure 6 and show a high matchof original and extrapolated functions even in most partsof the fine structure Some regions however show minordeviations inward extrapolation results in amplitude rip-ples and outward extrapolation yields small deviations forhigh frequencies This might be put down to the intersec-

78

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

(c) (d)

Figure 6 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 7 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulated at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

tions of the sphere of 30 cm radius with the torso of thedummy head 2 While the intersecting parts are removedsecondary sources close to the sphere still exist (caused byedge diffraction) which might create some errors A dis-continuity is clearly visible at 10 kHz in Figure 6 It orig-inates from a change of mesh size for BEM simulationsfor this dummy head as mentioned in section 31 As both

range extrapolation and simulation are performed for dis-crete frequency values the observed artifacts do not affectthe results negatively

Finally in Figure 7 measurements for dummy head 2are extrapolated in both directions and the results showa very good match between measured and extrapolatedHRTFs Here ripples not only occur for inward extrap-

79

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

360 azimuth with a step size of 10 in elevation and 5

in azimuthAll microphones had been equalized to compensate for

temporal phase and level differences The closest micro-phone position at full lateralization ie on the interauralaxis was defined as global time reference point Diffuse-field equalization filters as described in [36 37] were ap-plied to all measurements These filters are derived froma weighted average over all measurements for one micro-phone at a known distance The applied weighting coeffi-cients correspond to the Voronoi surfaces [30] associatedwith each measurement position After the diffuse-fieldequalization directional transfer functions (DTFs) wereobtained for further calculations

Dummy head 2 ndash Inward measurement The near-fieldHRTFs for dummy head 2 were measured by Lentz [38]in a semi-anechoic chamber (11 m times 6 m times 5 m) at ITAAachen This anechoic room provides a lower cut-off fre-quency of approx 100 Hz Similarly to the setup used atIRCAM a computerized positioning system consisting ofa turntable and a rotary arm for a moving sound sourcewere used that allows to measure HRTFs in the proximalregion of the head

The dummy head was mounted and aligned on theturntable and a pair of microphones was positioned at theentrance of the ear canals using custom made earmoldshells The entire measurement equipment was calibratedand referenced to the measured sound pressure of a mi-crophone in absence of the dummy head yielding the di-rectional transfer functions (DTFs) Due to the solid floorin the semi-anechoic room a strong first ground reflectionoccurred which was eliminated by extracting the first partof the measured impulse response

An equiangular spatial sampling grid with an angularresolution of 5 in elevation and azimuthal direction wasapplied which results in 2664 measurement positions onthe full sphere In practice for source distances close tothe head only partial sphere measurements could be ob-tained Thus the coverage of the HRTF data varies withthe measurement distance Whereas for distances at 20 and30 cm only the upper hemispherical HRTF data could beobtained the measurements at distances 40 50 75 100and 200 cm cover larger parts of the sphere encompassingall elevations from +90 (above the head) to -45 -55-75 -75 and -90 respectively

A more detailed description of the measurement setupcan be found in [38]

4 Analysis of the results

In this section the results of the proposed methods aregiven First the correlation analysis is used to show thematch of original and extrapolated HRTFs as functions offrequency This provides the possibility to study the ef-fect of regularization on partially defined data Secondthe HRTFs in the horizontal plane are plotted over fre-quency which allows to observe the fine structure of origi-nal and calculated functions in the spatial domain Finally

1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 2 Correlation between measured HRTFs and extrapo-lation results from measurement data obtained at a differentrange (dummy head 1) a Calculation without regularization(weighted least squares) b Calculation using decomposition-order dependent regularization with ε = 10minus8

the given calculation using the spherical wave spectrum iscompared to alternative near-field compensation methodsfound in literature

41 Correlation analysis of the HRTF extrapolation

The normalized correlation function C(f g) as defined inequation (12) is used to evaluate the range extrapolationof HRTFs in the spherical wave spectral domain Therebythe lower bound C = 0 and the upper bound C = 1 cor-respond to a total mismatch and a perfect match of thesimulated and extrapolated data respectively

Measurement data The correlation over frequency formeasurement data of dummy head 1 is depicted in Fig-ure 2 As mentioned earlier diffuse field normalizationwas applied to all data sets before correlation the resultsthus refer to the directional transfer functions (DTFs) Asa result of missing measurement data around the southpole ie the polar gap the weighted least squares ap-proach fails To obtain accurate range extrapolation data inboth outward and inward directions regularization meth-ods have been applied The regularization matrix is givenin equation (8) and the regularization parameter was set toε = 10minus8 In doing so the high frequency part of the ana-lyzed HRTFs could be more accurately reproduced thanwhen using ε = 10minus6 as has been suggested in [25]As already mentioned in Sec 21 the maximum sphericalharmonic order was truncated to N = krmin see also[25] rmin was set to 30 cm and thus encloses all scatteringsources

The correlation measure shows similar results for boththe inward and the outward extrapolation of HRTFs Therelatively high correlation does not reflect differences inthe fine structure of the HRTFs and their dependencies onthe direction of range extrapolation This will be discussedin more detail in Sec 42 The correlation curves for bothinward and outward extrapolation follow the same trend

76

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

and show a drop above 6 kHz which might be caused bymeasurement uncertainties

BEM simulation data Numerical simulations allow toanalyze the HRTF range extrapolation and interpolationmethods without the effects of measurement noise For thisreason the BEM method described in Sec 31 was appliedto dummy head 1 The simulation results were truncatedto elevations from+90 (north pole) to -30 and thus coverthe same portion of the sphere as the measured HRTFs

Figure 3 depicts the correlation between range extrapo-lated and BEM simulated HRTFs derived from complete(full sphere) and incomplete (partial sphere) data sets andfor different inversion methods The range extrapolatedHRTFs are in a good agreement with the BEM simulationresults and show in general slightly higher accuracy fornear-field to far-field range extrapolation than vice versa

It can be seen that the correlation monotonically de-creases with increasing frequencies up to 180 Hz This isbecause the spherical harmonic of order zero (monopole)fails to model the spatial variations of the HRTF dataAbove this frequency the maximum order is set to one(monopole plus dipole) and the extrapolated HRTFs areagain well correlated with the BEM simulations Thiseffect can be minimized by introducing smooth (order-weighted) transitions or by increasing the maximum orderin this frequency region However due to the high corre-lation values (above 095) this has not been investigatedfurther in this study as it is unlikely to cause significantaudible artifacts

The correlation curves in Figures 3b and c illustrate theimportance of regularization for data from measurementsover an incomplete portion of the sphere Figure 3b clearlyshows that weighted least squares inversion fails for thepartial sphere at frequencies above 15 kHz with the cor-relation decreasing rapidly for higher frequencies In Fig-ure 3c Tikhonov regularization with a regularization pa-rameter ε = 10minus4 was applied and the extrapolated HRTFsare well correlated with the BEM simulated data over theentire frequency range It is important to note that the BEMsimulations were limited to frequencies up to 6 kHz How-ever measurement results for the same dummy head asdepicted in Figure 2 show a good agreement of range ex-trapolated and measured HRTFs over almost the entire fre-quency range of human hearing (up to 16 kHz) when reg-ularization is applied

42 Azimuthal frequency plots for outward and in-ward range extrapolation

In this section the results of inward and outward range ex-trapolation are compared graphically Only the azimuthalvariation (horizontal plane of the HRTFs) is depicted asa magnitude plot over frequency Hereby the radiation isnormalized to the average (omnidirectional) radiation foreach frequency each plot thus denotes the deviation fromthe average as level values in decibels

For all dummy heads the results for the left ear were an-alyzed each of the plots in Figures 4 to 8 thus show their

20 40 60 100 200 400 1k 2k 4k 6k09

095

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

(c)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 3 Correlation between extrapolated HRTFs and corre-sponding BEM simulation results for dummy head 1 at dis-tances 200 cm (solid line) and 30 cm (dashed line) The range ex-trapolation is calculated from BEM simulated HRTF data whichcovers (a) the full sphere or (b)ndash(c) only a partial sphere witha polar gap below minus30 elevation (with and without regulariza-tion) Thus ldquo30 cmrarr200 cm denotes range extrapolation fromradius 30 cm to 200 cm compared to BEM simulation results atradius 200 cm a Full sphere without regularization (weightedleast squares) b Partial sphere without regularization (weightedleast squares) c Partial sphere standard Tikhonov regularizationwith ε = 10minus4

highest levels on the left-hand sides The dynamic rangewas chosen to give enough headroom for amplifications(+10 dB) and allow 30 dB below the average radiation tovisualize the deviations for the given dynamic range De-pending on the type of data the frequency axis was cho-sen to show all valid spectral information of the data Allplots are normalized and thus no information about ab-solute level mismatches can be derived from the figuresHowever level attenuation is a simple process whereasthe fine structure of the HRTF is most meaningful so nor-malized figures help visualizing these small differences

Dummy head 1 The BEM-simulated and range extrap-olated HRTFs show a general good match cf Figure 4The inward range extrapolation results in stronger rip-ples cf Figure 4c which is caused by the amplificationof higher orders in the spherical wave spectrum In con-trast outward range extrapolation attenuates higher ordersand thus always results in smoother functions

As no significant portions of the data coverage are miss-ing regularization is not required Nevertheless a smallregularization paramater of ε = 10minus8 was set in order tobe consistent with the other simulations in this section

As mentioned above BEM simulations have been lim-ited to frequencies below 6 kHz To study range extrapo-

77

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 4 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 5 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF measured at 30 cm b HRTF measured at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

lation for higher frequencies measured HRTFs were usedThe results clearly show that inward extrapolation createsnoisy artifacts whereas outward extrapolation results innoticeably smoother functions In general the data matchis good even if clearly visible deviations exist These devi-ations are most-likely caused by measurement uncertain-ties

Dummy head 2 The BEM simulation results of dummyhead 2 are depicted in Figure 6 and show a high matchof original and extrapolated functions even in most partsof the fine structure Some regions however show minordeviations inward extrapolation results in amplitude rip-ples and outward extrapolation yields small deviations forhigh frequencies This might be put down to the intersec-

78

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

(c) (d)

Figure 6 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 7 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulated at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

tions of the sphere of 30 cm radius with the torso of thedummy head 2 While the intersecting parts are removedsecondary sources close to the sphere still exist (caused byedge diffraction) which might create some errors A dis-continuity is clearly visible at 10 kHz in Figure 6 It orig-inates from a change of mesh size for BEM simulationsfor this dummy head as mentioned in section 31 As both

range extrapolation and simulation are performed for dis-crete frequency values the observed artifacts do not affectthe results negatively

Finally in Figure 7 measurements for dummy head 2are extrapolated in both directions and the results showa very good match between measured and extrapolatedHRTFs Here ripples not only occur for inward extrap-

79

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

and show a drop above 6 kHz which might be caused bymeasurement uncertainties

BEM simulation data Numerical simulations allow toanalyze the HRTF range extrapolation and interpolationmethods without the effects of measurement noise For thisreason the BEM method described in Sec 31 was appliedto dummy head 1 The simulation results were truncatedto elevations from+90 (north pole) to -30 and thus coverthe same portion of the sphere as the measured HRTFs

Figure 3 depicts the correlation between range extrapo-lated and BEM simulated HRTFs derived from complete(full sphere) and incomplete (partial sphere) data sets andfor different inversion methods The range extrapolatedHRTFs are in a good agreement with the BEM simulationresults and show in general slightly higher accuracy fornear-field to far-field range extrapolation than vice versa

It can be seen that the correlation monotonically de-creases with increasing frequencies up to 180 Hz This isbecause the spherical harmonic of order zero (monopole)fails to model the spatial variations of the HRTF dataAbove this frequency the maximum order is set to one(monopole plus dipole) and the extrapolated HRTFs areagain well correlated with the BEM simulations Thiseffect can be minimized by introducing smooth (order-weighted) transitions or by increasing the maximum orderin this frequency region However due to the high corre-lation values (above 095) this has not been investigatedfurther in this study as it is unlikely to cause significantaudible artifacts

The correlation curves in Figures 3b and c illustrate theimportance of regularization for data from measurementsover an incomplete portion of the sphere Figure 3b clearlyshows that weighted least squares inversion fails for thepartial sphere at frequencies above 15 kHz with the cor-relation decreasing rapidly for higher frequencies In Fig-ure 3c Tikhonov regularization with a regularization pa-rameter ε = 10minus4 was applied and the extrapolated HRTFsare well correlated with the BEM simulated data over theentire frequency range It is important to note that the BEMsimulations were limited to frequencies up to 6 kHz How-ever measurement results for the same dummy head asdepicted in Figure 2 show a good agreement of range ex-trapolated and measured HRTFs over almost the entire fre-quency range of human hearing (up to 16 kHz) when reg-ularization is applied

42 Azimuthal frequency plots for outward and in-ward range extrapolation

In this section the results of inward and outward range ex-trapolation are compared graphically Only the azimuthalvariation (horizontal plane of the HRTFs) is depicted asa magnitude plot over frequency Hereby the radiation isnormalized to the average (omnidirectional) radiation foreach frequency each plot thus denotes the deviation fromthe average as level values in decibels

For all dummy heads the results for the left ear were an-alyzed each of the plots in Figures 4 to 8 thus show their

20 40 60 100 200 400 1k 2k 4k 6k09

095

1

Frequency in Hz(a)

30cm rarr 200cm

200cm rarr 30cm

(b)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

(c)20 40 60 100 200 400 1k 2k 4k 6k

09

095

1

Frequency in Hz

30cm rarr 200cm

200cm rarr 30cm

Figure 3 Correlation between extrapolated HRTFs and corre-sponding BEM simulation results for dummy head 1 at dis-tances 200 cm (solid line) and 30 cm (dashed line) The range ex-trapolation is calculated from BEM simulated HRTF data whichcovers (a) the full sphere or (b)ndash(c) only a partial sphere witha polar gap below minus30 elevation (with and without regulariza-tion) Thus ldquo30 cmrarr200 cm denotes range extrapolation fromradius 30 cm to 200 cm compared to BEM simulation results atradius 200 cm a Full sphere without regularization (weightedleast squares) b Partial sphere without regularization (weightedleast squares) c Partial sphere standard Tikhonov regularizationwith ε = 10minus4

highest levels on the left-hand sides The dynamic rangewas chosen to give enough headroom for amplifications(+10 dB) and allow 30 dB below the average radiation tovisualize the deviations for the given dynamic range De-pending on the type of data the frequency axis was cho-sen to show all valid spectral information of the data Allplots are normalized and thus no information about ab-solute level mismatches can be derived from the figuresHowever level attenuation is a simple process whereasthe fine structure of the HRTF is most meaningful so nor-malized figures help visualizing these small differences

Dummy head 1 The BEM-simulated and range extrap-olated HRTFs show a general good match cf Figure 4The inward range extrapolation results in stronger rip-ples cf Figure 4c which is caused by the amplificationof higher orders in the spherical wave spectrum In con-trast outward range extrapolation attenuates higher ordersand thus always results in smoother functions

As no significant portions of the data coverage are miss-ing regularization is not required Nevertheless a smallregularization paramater of ε = 10minus8 was set in order tobe consistent with the other simulations in this section

As mentioned above BEM simulations have been lim-ited to frequencies below 6 kHz To study range extrapo-

77

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 4 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 5 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF measured at 30 cm b HRTF measured at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

lation for higher frequencies measured HRTFs were usedThe results clearly show that inward extrapolation createsnoisy artifacts whereas outward extrapolation results innoticeably smoother functions In general the data matchis good even if clearly visible deviations exist These devi-ations are most-likely caused by measurement uncertain-ties

Dummy head 2 The BEM simulation results of dummyhead 2 are depicted in Figure 6 and show a high matchof original and extrapolated functions even in most partsof the fine structure Some regions however show minordeviations inward extrapolation results in amplitude rip-ples and outward extrapolation yields small deviations forhigh frequencies This might be put down to the intersec-

78

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

(c) (d)

Figure 6 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 7 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulated at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

tions of the sphere of 30 cm radius with the torso of thedummy head 2 While the intersecting parts are removedsecondary sources close to the sphere still exist (caused byedge diffraction) which might create some errors A dis-continuity is clearly visible at 10 kHz in Figure 6 It orig-inates from a change of mesh size for BEM simulationsfor this dummy head as mentioned in section 31 As both

range extrapolation and simulation are performed for dis-crete frequency values the observed artifacts do not affectthe results negatively

Finally in Figure 7 measurements for dummy head 2are extrapolated in both directions and the results showa very good match between measured and extrapolatedHRTFs Here ripples not only occur for inward extrap-

79

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 4 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 5 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 1) plotted in the horizontal plane over frequency a HRTF measured at 30 cm b HRTF measured at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

lation for higher frequencies measured HRTFs were usedThe results clearly show that inward extrapolation createsnoisy artifacts whereas outward extrapolation results innoticeably smoother functions In general the data matchis good even if clearly visible deviations exist These devi-ations are most-likely caused by measurement uncertain-ties

Dummy head 2 The BEM simulation results of dummyhead 2 are depicted in Figure 6 and show a high matchof original and extrapolated functions even in most partsof the fine structure Some regions however show minordeviations inward extrapolation results in amplitude rip-ples and outward extrapolation yields small deviations forhigh frequencies This might be put down to the intersec-

78

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

(c) (d)

Figure 6 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 7 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulated at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

tions of the sphere of 30 cm radius with the torso of thedummy head 2 While the intersecting parts are removedsecondary sources close to the sphere still exist (caused byedge diffraction) which might create some errors A dis-continuity is clearly visible at 10 kHz in Figure 6 It orig-inates from a change of mesh size for BEM simulationsfor this dummy head as mentioned in section 31 As both

range extrapolation and simulation are performed for dis-crete frequency values the observed artifacts do not affectthe results negatively

Finally in Figure 7 measurements for dummy head 2are extrapolated in both directions and the results showa very good match between measured and extrapolatedHRTFs Here ripples not only occur for inward extrap-

79

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

(a) (b)

(c) (d)

Figure 6 Comparison of simulated data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalizedlevels of the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulatedat 200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

(a) (b)

(c) (d)

Figure 7 Comparison of measured data and extrapolated results using order-dependent regularization with ε = 10minus8 Normalized levelsof the HRTFs (dummy head 2) plotted in the horizontal plane over frequency a HRTF simulated at 30 cm b HRTF simulated at200 cm c Inward reconstruction of (a) using extrapolation of (b) d Outward reconstruction of (b) using extrapolation of (a)

tions of the sphere of 30 cm radius with the torso of thedummy head 2 While the intersecting parts are removedsecondary sources close to the sphere still exist (caused byedge diffraction) which might create some errors A dis-continuity is clearly visible at 10 kHz in Figure 6 It orig-inates from a change of mesh size for BEM simulationsfor this dummy head as mentioned in section 31 As both

range extrapolation and simulation are performed for dis-crete frequency values the observed artifacts do not affectthe results negatively

Finally in Figure 7 measurements for dummy head 2are extrapolated in both directions and the results showa very good match between measured and extrapolatedHRTFs Here ripples not only occur for inward extrap-

79

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

(a) (b)

(c) (d)

Figure 8 Comparison with alternative near-field compensation techniques Normalized levels of the HRTFs for dummy head 2 plot-ted in the horizontal plane over frequency a Original HRTF simulated at 30 cm b Inward reconstruction using the spherical wavespectrum c Predicted HRTF using the compensation filter method d Predicted HRTF using the cross-ear selection method

olation but also for outward extrapolation These ripplescan be also observed in the original data set measured inthe vicinity of the head at a radius of 30 cm

43 Comparison with alternative methods for near-field compensation

In this section the spherical acoustic range extrapolationof HRTFs is compared with other methods proposed inliterature such as the inverse distance attenuation coeffi-cient near-field compensation filters [27] and the selec-tion of HRTFs [24] All methods were applied to BEMsimulations of dummy head 2 for calculating the near-field HRTFs at radius 30 cm from data on a spherical sur-face with 2 m radius The BEM simulation allows to evalu-ate and compare the different range extrapolation methodswithout the influence of measurement noise

The first method which is herein referred to as dis-tance attenuation method applies a simple frequency-independent distance attenuation factor g(ri) = rir0The second method referred to as compensation filtermethod uses distance compensation filters [24] derivedfrom a spherical head model evaluated at distances r1 andr0 [27] The third method referred to as cross-ear selec-tion method extends the near-field compensation filter ap-proach by monaural gain corrections and a geometricallyderived selection of HRTFs depending on the relative po-sition of the source to the respective ear It thus modelsthe angular deviations of near-field HRTFs compared tothe corresponding far-field HRTFs for selecting the mostappropriate filter sets

The latter two range extrapolations are compared to theBEM simulated HRTFs at 30 cm radius see Figure 8a and

20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz(a)

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

(b)20 40 60 100 200 400 1k 2k 4k 6k 10k

06

08

1

Frequency in Hz

distance attenuation

compensation filter

cross-ear selection

spherical wave spectrum

Figure 9 Correlation (calculated in horizontal plane only) be-tween an extrapolated and simulated set of HRTFs using differ-ent extrapolation methods (dummy head 2) a Inward extrapo-lation b Outward extrapolation

the synthesized HRTFs using the spherical wave spectrummethod with regularization (ε = 10minus8) at the same dis-tance see Figure 8b One can see in Figure 8b that theoverall match of HRTF data is high indicating that thespherical harmonics method performs best for inward pre-diction of near-field HRTFs Compared to the other rangeextrapolation methods it shows stronger magnitude varia-tions in the fine structure of the HRTF see also section 42

In Figure 9 the correlations between orginal and extrap-olated HRTFs are plotted for the different range extrapola-tion methods As the cross-ear method generally modifiesthe elevation of the used points the normalized correla-

80

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

Pollow et al HRTFs for arbitrary field points ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

tions are evaluated on the horizontal plane only (wherethere is no change in elevation) Except for the artifactsin the low frequency range the proposed method usingthe spherical wave spectrum clearly shows the best matchwith the original HRTFs Due to normalization the ripplesover frequency cannot be observed in these plots Withregard to the correlation curves both extrapolation direc-tions show a similar performance

5 Conclusions

In this paper the calculation of head-related transfer func-tions for arbitrary field points has been investigated Thespherical acoustic method which provides efficient algo-rithms for both interpolation and range extrapolation ofHRTFs was compared to other methods proposed in liter-ature using the correlation of the original and extrapolatedfunctions as a measure of their similarity For this studyHRTFs of two different dummy heads were measured in ananechoic chamber and calculated using boundary-elementmethod (BEM) simulations

HRTFs are typically obtained for a large number of spa-tial positions on a surrounding sphere and can thus be rep-resented in the spherical wave spectral domain The rangeextrapolation of HRTFs can therefore be performed by ap-plying the radial wave propagation terms to the sphericalharmonic expansion coefficients For incomplete data cov-erage the spherical harmonic transform is not always feasi-ble However using regularization allows to perform inter-polation and extrapolation accurately up to high frequen-cies also for data that does not cover the full sphere

The spherical acoustic method for range extrapolationis more precise than alternative methods proposed in lit-erature It turned out that the extrapolated HRTFs are invery good agreement with BEM simulation results andin sufficient agreement with acoustic measurements in theproximal region of the head The inward range extrapo-lation ie from far-field to near-field is strongly affectedby small frequency-dependent deviations presumably dueto the high amplification of high-order spherical harmoniccoefficients Contrary to what is expected from theory out-ward range extrapolation was in general not more accuratethan inward range extrapolation A more thorough com-parison of the performance of inward and outward extrap-olation is suggested for further research

Using spherical harmonic decomposition of HRTFs al-lows to calculate appropriate function values thereof at anyposition in the three-dimensional space around the headThis is of particular interest for interactive virtual-realityapplications To further reduce the computational load forthe use in real-time applications the used Hankel filterscan be pre-calculated off-line for different source posi-tions and further combined with efficient nearest-neighborsearch algorithms In the future subjective listening ex-periments with individually measured HRTFs should becarried out to investigate the perceptual impact of the ob-served artifacts

Acknowledgment

The authors are grateful to Tobias Lentz for providing themeasured near-field HRTF data of dummy head 2 and toRobin Lenogue for help with experimental procedures atIRCAM This research was supported in part (IRCAM)by the project ldquoSAME ndash Sound and Music for EveryoneEveryday Everywhere Everyway funded by the Euro-pean Commission under the 7th Framework ProgrammeTheme ICT-200715 Networked media

References

[1] J Blauert Spatial hearing The psychophysics of humansound localization The MIT Press Boston USA 1997

[2] J C Middlebrooks D M Green Sound localization byhuman listeners Annual Review of Psychology 42 (1991)135ndash159

[3] F L Wightman D J Kistler Headphone simulation offree-field listening I Stimulus synthesis J Acoust SocAm 85 (1989) 858

[4] E A G Shaw R Teranishi Sound pressure generated in anexternal-ear replica and real human ears by a nearby pointsource J Acoust Soc Am 44 (1968) 240ndash249

[5] W G Gardner K Martin HRTF measurement of aKEMAR J Acoust Soc Am 97 (1995) 3907ndash3908

[6] V R Algazi R O Duda D M Thompson C AvendanoThe CIPIC HRTF database IEEE WASPAA New PaltzUSA 2001 99ndash102

[7] P Majdak P Balazs B Laback Multiple exponentialsweep method for fast measurement of head-related trans-fer functions J Audio Eng Soc 55 (2007) 623

[8] Y Kahana P A Nelson M Petyt S Choi Numericalmodeling of the transfer functions of a dummy head andof the external ear 16th Int Conf Audio Eng SocRovaniemi Finland 1999 330ndash345

[9] B F G Katz Boundary element method calculation of in-dividual head-related transfer function I Rigid model cal-culation J Acoust Soc Am 110 (2001) 2440

[10] T Walsh L Demkowicz R Charles Boundary elementmodeling of the external human auditory system J AcoustSoc Am 115 (2004) 1033ndash1043

[11] M Otani S Ise Fast calculation system specialized forhead-related transfer function based on boundary elementmethod J Acoust Soc Am 119 (2006) 2589ndash2598

[12] N A Gumerov A E OrsquoDonovan R Duraiswami D NZotkin Computation of the head-related transfer functionvia the fast multipole accelerated boundary element methodand its spherical harmonic representation J Acoust SocAm 127 (2010) 370ndash386

[13] E M Wenzel M Arruda D J Kistler F L WightmanLocalization using nonindividualized head-related transferfunctions J Acoust Soc Am 94 (1993) 111

[14] E H A Langendijk A W Bronkhorst Fidelity of three-dimensional-sound reproduction using a virtual auditorydisplay J Acoust Soc Am 107 (2000) 528ndash537

[15] V Larcher J Jot Techniques drsquointerpolation de filtresaudio-numeacuteriques Application agrave la reproduction spatialedes sons sur eacutecouteurs Congres Franccedilais drsquoAcoustique(CFA) Marseille France 1997

[16] D J Kistler F L Wightman A model of head-relatedtransfer functions based on principal components analysisand minimum-phase reconstruction J Acoust Soc Am91 (1992) 1637ndash1647

81

ACTA ACUSTICA UNITED WITH ACUSTICA Pollow et al HRTFs for arbitrary field pointsVol 98 (2012)

[17] J Chen B D Van Veen K E Hecox A spatial feature ex-traction and regularization model for the head-related trans-fer function J Acoust Soc Am 97 (1995) 439ndash452

[18] M J Evans J A S Angus A I Tew Analyzing head-related transfer function measurements using surface spher-ical harmonics J Acoust Soc Am 104 (1998) 2400ndash2411

[19] D N Zotkin R Duraiswami N A Gumerov RegularizedHRTF fitting using spherical harmonics IEEE WASPAANew Paltz USA 2009

[20] P Zahorik D S Brungart A W Bronkhorst Auditory dis-tance perception in humans A summary of past and presentresearch Acta Acustica united with Acustica 91 (2005)409ndash420

[21] D S Brungart W M Rabinowitz Auditory localization ofnearby sources head-related transfer functions J AcoustSoc Am 106 (1999) 1465ndash1479

[22] T Lentz I Assenmacher M Vorlaumlnder T Kuhlen Precisenear-to-head acoustics with binaural synthesis Journal ofVirtual Reality and Broadcasting 3 (2006)

[23] C Borszlig An improved parametric model for the design ofvirtual acoustics and its applications PhD dissertationRuhr-Universitaumlt Bochum Germany 2010

[24] D Romblom B Cook Near-field compensation for hrtfprocessing 125th Conv Audio Eng Soc San FranciscoUSA 2008 no 7611

[25] R Duraiswami D N Zotkin N A Gumerov Interpola-tion and range extrapolation of HRTFs IEEE ICASSPMontreal Canada 2004 45ndash48

[26] W Zhang T D Abhayapala R A Kennedy R Du-raiswami Modal expansion of HRTFs Continuous repre-sentation in frequency-range-angle ICASSP Los Alami-tos USA IEEE Computer Society 2009 285ndash288

[27] R O Duda W L Martens Range dependence of the re-sponse of a spherical head model J Acoust Soc Am 104(1998) 3048ndash3058

[28] E G Williams Fourier acoustics Sound radiation andnearfield acoustical holography Academic Press 1999

[29] A Neumaier Solving ill-conditioned and singular linearsystems A tutorial on regularization SIAM Review 40(1998) 636

[30] H S Na C N Lee O Cheong Voronoi diagrams on thesphere Computational Geometry 23 (2002) 183ndash194

[31] P J Kostelec D N Rockmore FFTs on the rotation groupJournal of Fourier Analysis and Applications 14 (2008)145ndash179

[32] M Pollow Variable directivity of dodecahedron loud-speakers Masterrsquos thesis Institute of Technical AcousticsRWTH Aachen University 2007

[33] A Schmitz Ein neues digitales KunstkopfmeszligsystemAcustica 81 (1995) 416ndash420

[34] D N Zotkin R Duraiswami E Grassi N A GumerovFast head-related transfer function measurement via reci-procity J Acoust Soc Am 120 (2006) 2202ndash2215

[35] A Farina Simultaneous measurement of impulse responseand distortion with a swept-sine technique 108th ConvAudio Eng Soc Paris France 2000 no 5093

[36] V Larcher G Vandernoot J M Jot Equalization methodsin binaural technology 105th Conv Audio Eng Soc no4858 San Francisco USA 1998 26ndash29

[37] V Larcher Techniques de spatialisation des sons pour lareacutealiteacute virtuelle PhD dissertation Universiteacute de Paris VIFrance 2001

[38] T Lentz Binaural technology for virtual reality PhD dis-sertation Institute of Technical Acoustics RWTH AachenUniversity Germany 2007

82